Well pushed forward from the start with Dave Clarkson firing over from a Scott McDonald backheel in the second minute.
Falkirk were busy without creating any threat in the opening 30 minutes, but they then scored with their first shot.
Alan Gow released Anthony Stokes on the left, he turned Paul Quinn and fired past Colin Meldrum from an acute angle.
Well were handed a lifeline when they were awarded a penalty when the ref spotted the ball strike Marc Twaddle's arm and Richie Foran made no mistake from the spot.
Minutes after the re-start the home side were awarded a second penalty when Keith Lasley was felled by Kenny Milne and Foran once again made no mistake.
Falkirk equalised when Russell Latapy stroked home an 18-yard shot from a Stokes lay-off with 20 minutes left.
They were level for only five minutes before McDonald fired a shot high past Jeroen Lambers and soon after Stokes was shown a second yellow card when he appeared to dive in an attempt to win a penalty.
Well put the icing on their cake with eight minutes left when Calum Elliot scored a fourth.