The Galaxy S6 Edge uses the feature to provide a quick way to stay in touch with select contacts, and to alert owners to important information.
It will be sold for a higher price than the standard S6, which otherwise has the same specifications.
Samsung lost market share to Apple and others after the S5 sold fewer copies than its predecessor in many countries.
Its replacements were unveiled in Barcelona, ahead of the start of Mobile World Congress. They go on sale on 10 April in 20 countries, including the UK.
The South Korean firm said it had set out to address past "missteps", and had codenamed the devices "project zero" to reflect the need for a rethink.
Changes include making the TouchWiz user interface simpler to use by cutting the number of pop-up messages and introducing a metal frame and glass back instead of the plastic styling of earlier models.
The redesign has, however, meant some features have had to be jettisoned: the phones are not water-resistant, they do not have a microSD slot for extra storage and their backs cannot be removed to change their batteries.
Even so, one expert praised the company's choices.
"Samsung has clearly listened to feedback - not just for the Galaxy S5, which just missed the mark last year costing the company greatly - but also the S4, which was a product that tried to cram in every piece of technology it could find," said Ben Wood, head of research at the CCS Insight tech consultancy.
"The software has been made into a much more crisp and clear experience, the design of the product has clean lines and looks very nice, and the marketing campaign is expected to only pinpoint three things - and that's certainly something that had been missing from Samsung's products for quite some time."
Others were more critical. Rob Kerr, from the price comparison site uswitch.com, declared: "The dual-curved display just seems too gimmicky, too niche, to really be a crowd pleaser."
The three topics Samsung is focusing on are:
The S6 Edge uses its curves to provide a couple of services.
The first is called People Edge, which provides a quick way to bring up calls, texts and other messages from five acquaintances of the owner's choosing. Each person is assigned a different colour, which the phone's edge flashes when it rings, providing a hint of the caller's identity even if the device is face down.
The second is Information Stream, which displays the time, weather and selected notifications on the curved part.
It appears Samsung has deliberately kept the functionality more basic than on its Galaxy Note Edge - a larger handset whose screen curves only around one of its sides. The older phone uses the extra space to both run apps of its its own and to add controls to other apps.
The S6 phone's front camera has been upgraded to five megapixels, while the rear one stays at 16MP.
Both gain from a wider aperture, which should improve their ability to take photos and videos in low light conditions.
Samsung has also taken steps to make the camera quicker to use, saying it now takes less than a second to double tap the home key and snap a shot.
The firm is claiming a "world first" by embedding support for both the PMA and WPC's Qi wireless charging standards.
The Android-powered handsets are also faster to recharge, and should return to 50% battery strength within half an hour of being plugged in.
Samsung said that this was half the time it would take to charge an iPhone 6.
Other improvements over the S5 include:
The firm also revealed it had developed a new version of its Gear VR virtual reality kit that uses the S6 models as a screen.
A lot is resting on the phones' appeal.
Samsung's head of mobile design was moved from his role in May after criticism of the Galaxy S5's styling, and last week Samsung Electronics announced it would freeze its workers' wages for the first time in six years following a drop in its annual profits.
Samsung is still the best-selling smartphone maker.
But it barely saw its total smartphone shipments grow in 2014. By contrast, Apple's decision to make its iPhone 6 models larger and the continued rise of China's Xiaomi, Lenovo and Huawei saw those firms report much bigger gains.
As a result, Samsung's market share slipped from 31% in 2013 to 24.5% last year, according to research firm IDC.
The price of the new handsets has yet to be announced, but Mr Wood thought the S6 Edge in particular could mark a turning point.
"In a sea of sameness where nearly every smartphone is a rectangular touchscreen with a camera in the back, this will stand out," he said.
"It may be outside many people's budgets, but it will still act as a signature product that attracts people to the rest of Samsung's range."
1. Samsung Galaxy
2. Samsung Galaxy S
3. Samsung Galaxy S2
4. Samsung Galaxy S3
5. Samsung Galaxy S4
6. Samsung Galaxy S5
7. Samsung Galaxy S6
Samsung’s first Android-powered phone ran version 1.5 (Cupcake) of Google’s mobile operating system.
It had a 3.2in (8.1cm) screen and was the first Android phone to include a 3.5mm headphone jack, allowing normal headphones to be easily connected.
The first of Samsung’s S-branded series had a 4in (10.2cm) screen, ran Android 2.1 (Eclair) and could shoot video in 720p high definition.
A 4G variant of the device attracted the ire of Apple’s lawyers, and became part of a high-profile patent case fought by the two rivals.
The S2 was marketed as the world’s thinnest smartphone, measuring less than 8.5mm (0.33in) thick.
It had a 4.3in (10.9cm) screen and was powered by Android 2.3 (Gingerbread).
Samsung promoted the “zero lag” shutter speed of its main camera as being suitable for snapping fast-moving objects.
The S3 used eye-tracking software to detect when it was being looked at, keeping its screen bright until its owner turned away.
It had a 4.8in (12.cm) screen and ran Android 4.0.4 (Ice Cream Sandwich).
The phone also introduced S-Beam, a way to send files and contacts between Samsung handsets.
The S4 bordered on “phablet” status with a 5in (12.7cm) touchscreen, which could be used while wearing gloves.
Running Android 4.2.2 (Jelly Bean) it introduced a dual-video capture mode, allowing users to record footage from both cameras simultaneously.
The S5 added dust and water resistance, promising to work after being submerged 3ft (0.9m) deep.
It came pre-installed with Android 4.4.2 (KitKat) and introduced a fingerprint scanner and heart rate monitor.
Samsung promoted its “ultra power saving mode”, which let the device eke out a day of life when its battery fell below 10%.
Samsung launched two models of its flagship phone – the Edge version had a screen that curved round its sides and was more expensive.
The phones had brighter, more detailed screens than before, and supported the firm’s new smart wallet service Samsung Pay.
However, their metal frames and glass-backed designs led to some of the S5’s features being jettisoned, including water resistance and the microSD slot.
It charges faster than the new iPhone. It has better cameras than the iPhone. The video quality is higher than the new iPhone.
And it doesn't bend.
That's the takeaway from Samsung's latest Unpacked event where it unveiled the next Galaxy phone, the S6.
Sleek and powerful, certainly - but it was a fairly bitter presentation. The firm took more than a few snipes at Apple, rather than leaving it to reviewers and users to draw their own conclusions about the Galaxy's many innovations .
The demonstrations of like-for-like pictures and video appeared to put the S6 streets ahead of the iPhone 6, but I imagine Apple would dispute the conditions.
Tellingly, there was no mention of China's Xiaomi - a company whose Android handsets have proved wildly popular in Asia, and which potentially presents a far bigger threat to Samsung's future.
You can follow Dave at MWC via his Twitter account @daveleeBBC.
Samsung will sell two versions of its next flagship phone, one of which has a screen that curves round its sides.