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.
Falkirk beat Morton to land their first win in six games, moving two points behind Hibernian in the Championship.