Jay Fulton, son of Greenock-born former Celtic and Hearts playmaker Stevie, gave the Bairns the lead with a freak goal in the ninth minute.
The midfielder fired over an inswinging cross from out on the left which evaded everyone, including Derek Gaston, and found the bottom right corner.
David Weatherston passed up two decent chances to extend the visitors' lead before Ton levelled things up just after the half hour.
Stephen Stirling picked up the ball 35 yards out and put Archie Campbell, who started up front with David Graham as Allan Moore returned to a 4-4-2 formation, in on goal with just the goalkeeper to beat.
The speedy frontman made no mistake, placing a neat finish past Michael McGovern with his left foot. The sides were level for just eight minutes, however.
Steven Pressley looks to have replaced departed frontman Farid El Alagui with another quality striker in Lyle Taylor. And it was the Englishman who netted what turned to be the winner with a stunning strike on 39 minutes.
McGovern thwarted Campbell with another impressive stop in the second half, but that was as good as it got for Ton, who are without a win in their first three league games and have won just one of five home fixtures in all competitions.
Morton boss Moore said: "It's a sore one. We've lost two goals - a fluke and a wonder strike. I think when we got back to 1-1 we were dominating the game and played some excellent stuff, but it's fine margins. I'm fed up saying it."
Morton (4-4-2): Gaston; Taggart, Reid ©, Naismith, Dyer; Wallace (Weatherson, 56), Stirling (Wilkie, 72), Bachirou, O'Brien (Hawke, 85); Graham, Campbell.
Subs not used: Halsman, Combe (gk).
Booked: O'Brien (54), Taggart (59), Graham (78).
Falkirk (4-4-1-1): McGovern ©; Duffie, Flynn, Smith, Kingsley; Weatherston (Haworth, 83), Murdoch, J. Fulton, Alston; Sibbald (White, 88); Taylor.
Subs not used: McGeever, D. Fulton, Bowman (gk).
Booked: Duffie (21)
Referee: Craig Thomson