The former Dundee United forward, who has endured a battle with a persistent hamstring complaint this season, settled an attritional contest after rising highest to meet an Aidan Nesbitt cross, looping a header beyond the dive of Danny Rogers.
While reeling in the Bairns was their immediate aim, Morton are now only seven points behind Hibernian with a game in hand over the capital club, and two fixtures against them in the next three weeks.
“Kudus has had a tough time with injury and you are always delighted when they can come back and make a key contribution,” said Morton manager Jim Duffy. “It was his birthday too, he’s just turned 24. It was the ideal day for him.
“Sometimes you do get those strange things in football, which go for you. More than that, it shows we have depth in key areas and, when everyone is fit and healthy, we have a strong squad.”
Falkirk will rue a super first half save from Derek Gaston, parrying a John Baird volley clear, and Nathan Austin’s late header which struck the inside of the post as they passed up the chance to move within four points of Hibs ahead of next weekend’s trip to Easter Road.
“Being the later kick off, you do hear the other results coming in, so there was maybe that little bit more to play for,” admitted Falkirk assistant James McDonaugh. “But you can’t dwell on what might have been, we just need to prepare, go to Easter Road and hopefully start another run.”