Bell-Drummond (80 not out) and Denly (78) shared an opening stand of 131 after being set 234 to win.
Alex Blake smashed two sixes in his 34 not out as Kent reached 236-5.
Earlier, Michael Carberry made 61 as he shared a first-wicket partnership of 80 with Jimmy Adams (47), but Hampshire collapsed to 233 all out.
Kent's slow left-armer Fabian Cowdrey, who only had five domestic one-day wickets before the match, helped to bring about the demise of Hampshire's innings with career-best figures of 3-32.
Bell-Drummond then missed out on surpassing his limited overs best of 83, after Blake took the attack to the bowlers and scored the majority of the runs in their half-century stand.
Kent captain Sam Northeast:
"If we can get the first powerplay right, do things better up top than we did today, then we can be a proper side in this format. The guys did so well to drag us back into things.
"The run chase went fairly smoothly thanks to Danny [Bell-Drummond], who again showed how his one-day game is maturing when opening up alongside an experienced campaigner like Joe [Denly].
"Those two laid the foundations and made life pretty comfortable for the rest of us."

Apple won a ban on the devices, claiming Samsung had copied its iPhone and iPad.
South Korean Samsung had been unable to sell products in Australia since July.
The two companies are engaged in legal battles in more than 10 countries since April accusing each other of infringing smartphone and tablet patents.
Last month, Samsung won an appeal against a temporary ban on sales.
However, Apple was awarded a stay of the orders. On Friday Apple's bid to extend the ban was turned down by the High Court.
The ruling is a rare win for Samsung as it allows the company to sell its Galaxy Tab 10.1 in time for the busy Christmas shopping season, a lucrative time for the industry.
The decision comes after a US court ruled against Apple's bid to ban Galaxy phones and tablets, another win for Samsung. Apple has appealed against that ruling.
However, the legal tussle in other markets is not going as well for Samsung.
The company said on Friday that a French court had turned down its request to ban sale of the latest iPhone in that country.

The fire started in a bin in a nearby underpass, according to a spokesperson for the Northern Ireland Fire and Rescue Service (NIFRS).
They said the fire broke through into a tunnel and set fire to the dental building, run by Queen's University.
The building sustained smoke damage and some damage to its pipework, but there are no reports of injuries.
The blaze was reported about 20:20 BST on Thursday.
The NIFRS sent at least four fire engines to the scene, along with a team who specialise in dealing with hazardous materials.
The fire has been extinguished but a cordon was put in place for public safety.

She was performing in Verona in Italy when she spotted a woman in the audience with a tripod and camera filming her.
Talking to the woman in the crowd, Adele said: "Can you stop filming me with the video camera because I'm really here in real life and you can enjoy it in real life, rather than through your camera.
"Can you take your tripod down, this isn't a DVD, it's a real show and I'd really like you to enjoy my show because there's lots of people outside who couldn't come in."
The encounter was filmed by another fan in the audience, who posted it on Twitter.
Video courtesy of Madreeeh.
Kent openers Daniel Bell-Drummond and Joe Denly both hit half-centuries to lead their side to a five-wicket win over Hampshire in the One-Day Cup.
[NEXT_CONCEPT]
Australia's High Court has lifted a ban on the sale of Samsung's Galaxy tablet in the country in time for the Christmas shopping season.
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A fire has damaged a dental building close to the Royal Victoria Hospital in Belfast.
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Adele has stopped a concert mid-way through to tell a fan to stop filming her.