A goal in each half from Russell Latapy and Alan Gow proved enough and United had skipper Lee Wilkie sent off for two second-half bookings.
Falkirk took just five minutes to make their mark. Carl Finnigan pounced on a poor header by Sean Dillon to lob the advancing Derek Stillie.
David McCracken made a valiant attempt to clear the ball off the line but Latapy had followed up to bundle the ball into the net.
United hardly had time to recover before Vitor Lima rifled a 20-yard effort narrowly over the top.
Tom Scobbie then swept over an inviting cross in front of Stillie but none of his team-mates had read his intentions. Gow then saw a shot blocked by Wilkie with United again exposed at the back.
United tried to hit back and Greg Cameron found David Robertson in the area but his half-hit shot struck Darren Barr and the chance was lost.
Craig Levein's side came more into things as the first half wore on but too often their long spells in possession came to nothing.
A great run from Craig Conway almost produced an equaliser after 25 minutes but Steven Robb prodded the ball straight at Kasper Schmeichel from just eight yards out.
Gow made it 2-0 six minutes into the second half when he took a pass from substitute Pedro Moutinho to drill the ball into the corner of the net.
Liam Craig was then denied by Stillie when he had time to pick his spot from just 10 yards.
Wilkie was booked in 60 minutes for a foul on Vitor Lima and was sent off 12 minutes later for a heavy challenge on Barr.