The teenager struck with just four minutes to go at Gayfield Park after an equalising goal from Charlie Telfer just before half time had cancelled out an early opener from the Lichties’ Thomas O’Brien.
The reward from the Bairns is an away trip to the cup holders Celtic.
With one eye on the upcoming congested fixture list, Lee Miller and David McCracken made a huge seven changes to the side that had won against Montrose on Saturday, with Conor Sammon being handed the captain’s armband.
A strong, swirling wind blew in the hosts’ favour in the first half and it was no surprise that it had a hand in the opening goal.
Arbroath skipper O’Brien launched a cross into the Falkirk box which was picked up by the wind and eluded keeper Robbie Mutch and landed in the net.
O’Brien then came close again – the defender has two clear headers at goal on 31 and 36 minutes to add to his team's scoreline but put both over the bar.
And it would prove costly as shortly before half time Falkirk grabbed an equaliser.
On 42 minutes Akeel Francis picked the ball up in the middle of the park and made a great run to the edge of the Arbroath box. He squared it to Charlie Telfer who placed it low into Derek Gaston’s net.
The wind died down in the second half and it was the hosts who came closest to taking the lead when a Jack Hamilton shot smacked the post on 67 minutes, but it was Falkirk who would grab the winner.
A Sammon attempt on goal was blocked and the ball rebounded to 17-year-old Fotheringham who struck a right-footed angled drive high into the net.