John Baird's cool finish from a tight angle proved to be the difference.
That lead was nearly added to when Will Vaulks hit the crossbar from 20 yards, then Morton's Declan McManus did the same from 30 yards.
The game ended on a low note with Falkirk's on-loan Norwich midfielder Conor McGrandles suffering a serious leg injury.
Apart from Ton midfielder Joe McKee almost scoring directly from a corner and Vaulks' looping effort at the other end, the opening exchanges at Cappielow matched the dull weather.
But the uncovered, wet Bairns fans had something to cheer about when Baird used all his know-how to net his 17th goal of the season in front of the away end.
The ball had been booted high into the air in Morton's box and Baird outmuscled home defender Conor Pepper as it dropped, before staying calm to place it beyond goalkeeper Derek Gaston.
Vaulks' technique was ideal as he watched a side-footed curler kiss the the top of the woodwork soon after.
Morton's McKee directed a header wide when well placed and on-loan Fleetwood Town striker McManus - a revelation for Ton last season - went closer with a dipping free-kick just prior to half-time.
Bobby Barr rifled in a low left-footed shot just wide of the far post and McManus tested Danny Rogers again as Morton pushed for the equaliser.
But clear contender for man-of-the-match Baird came closest to hitting the back of the net in the second 45 minutes, forcing Gaston to make a brilliant clawed save from his half-volley.
McGrandles - in his second spell at Falkirk - came off the worst in a crunching tackle with McKee near the end, for which the Morton player was booked.
It meant eight minutes of added time but the visitors held on to secure their first victory since 2 January.

The car bonnet would be sticky so that if it hit a person, they would stick to the car.
This would stop the them falling back onto the road after the collision, which could injure them even more.
Google's idea is aimed at driverless cars but could be used on regular cars too.
Don't worry, the sticky layer would be covered by a special coating, so flies wouldn't get stuck to the car too.
It is just an idea at the moment but it could be something we see on our roads in the future!

Mandy and Pete Garland, from Weston-super-Mare, said their home address, phone numbers and flight dates were shared by Thomson in an email.
Thomson apologised for the error.
The Information Commissioner's Office (ICO) said it had investigated and was taking no further action.
Mr and Mrs Garland were among 458 customers whose booking details were shared on 15 August.
The couple booked a two-week trip to the Dominican Republic in January but later received a balance reminder email with a spreadsheet attached which included details of almost 500 customers from "Scotland to Cornwall", Mr Garland said.
"My biggest worry was how many people on that list are going to be prepared to show those details?
"Anybody with a little bit of brains would be able to look at the list and go 'there's a pretty good chance that house is going to be empty for at least a week'.
"So my biggest concern was the security of the house."
A Thomson spokesman said the email was "identified very quickly and recalled".
The couple said they had received "no apology and no money back". They said rescheduling their holiday "was not an option" as their jobs did not give them that flexibility.
Falkirk beat Morton to land their first win in six games, moving two points behind Hibernian in the Championship.
[NEXT_CONCEPT]
Google's got a new idea to keep pedestrians safer on the roads - having sticky cars!
[NEXT_CONCEPT]
A couple whose personal details were revealed in a data breach by holiday company Thomson have cancelled their Caribbean trip over fears they could be burgled while away.