David Goodwillie set the ball rolling on the win with his ninth goal of the season - from the penalty spot - and his overall performance no doubt impressed Scotland Under-21 boss Billy Stark ahead of their clash with Azerbaijan.
A clean sheet for the United defence would also have been noted by the watching Craig Levein, after he named both Garry Kenneth and Paul Dixon in his squad for Scotland's friendly against Czech Republic on Wednesday.
The home side made three changes from the side who lost narrowly to Celtic for the clash with the rock-bottom Bairns.
Dixon was back from injury while Darren Dods and Goodwillie were drafted back into the starting line-up, with Sean Dillon dropped to the bench, Jon Daly injured and Andy Webster also absent from the squad.
Unsurprisingly, Falkirk were unchanged from the side who claimed a surprise win over Aberdeen last week, as new boss Steven Pressley searched for his second victory since replacing Eddie May at the helm.
The Tannadice side tried to stamp their authority on the game in the opening few minutes when Craig Conway whipped a cross in from the left for Damien Casalinuovo, but he was denied at the near post by goalkeeper Robert Olejnik.
Both players were involved again when Conway's free-kick into the box was nodded on by Casalinuovo for Goodwillie, only for the youngster to hook just over from close range.
The home side's pressure paid off with 17 minutes gone when Goodwillie was needlessly tripped in the box by Scotland Under-21 team-mate Tam Scobbie and referee Calum Murray immediately pointed to the spot.
Goodwillie took the penalty himself and, although the run-up was somewhat hesitant, managed to convert with a powerful effort past Olejnik.
Falkirk could have hit back straight away when Vitor Lima stung the palms of United keeper Dusan Pernis with a ferocious shot from just inside the area.
At the other end, Danny Swanson almost doubled the advantage when he tried to lob Olejnik from 18 yards only to see the effort dip just over the crossbar.
Instead, it was Gomis who grabbed United's second of the day with 29 minutes gone. A Goodwillie strike was blocked by the keeper and Conway rattled the rebound off the post before Gomis eventually lashed home with the third attempt.
Falkirk thought they would be thrown a lifeline after claiming a Brian McLean shot was handled by Mihael Kovacevic, but the referee ignored their pleas for a penalty.
The game could have been all but over as a contest a few minutes after the restart when Swanson was allowed to make an unchallenged run before laying off for Conway, but his shot was inadvertently blocked by team-mate Dods.
Goodwillie was then able to skip his way past several Falkirk players before attempting to drill home his second of the day but an alert Olejnik came to the rescue and tipped to safety at the near post to deny the striker.
The visitors could have hauled themselves back into the match when Dixon slipped and allowed Lima to deliver a cross from the right to the feet of Enoch Showunmi, but the big hitman fired straight at the keeper.
Goodwillie then squandered a great chance when he miskicked in front of goal before ending up on his backside as the Falkirk defence gratefully hooked to safety.
It was Gomis who had the final say when he claimed his brace with just three minutes to go, guiding into the back of the net after being set up by substitute Dale Hilson.
The win, combined with Hibs' draw against St Johnstone, allowed United to move into the coveted third spot in the SPL behind the Old Firm.