Replays showed that the challenge was two yards outside of the area but top scorer John Baird stepped up and made no mistake to put the hosts ahead.
Mark Warburton's side looked uncomfortable in the early stages but managed to restore parity against the run of play on 16 minutes when on-loan Spurs winger Nathan Oduwa found Billie McKay at the back post.
Falkirk should have been back in front 10 minutes later but for an error of judgment from Baird. Seventeen-year-old Kevin O'Hara's strike was destined past Wes Foderingham in the Gers goal but an offside Baird collected the ball on the line for a tap-in.
After the restart, the visitors fell flat again and were punished for their lack of creativity as the hour mark approached when Vaulks shot low and hard through a busy box to restore Falkirk's lead.
Moments later it was almost three when a bending shot from O'Hara was denied by Foderingham, who showed good reaction to dart left and get his fingertips to the ball.
Warburton brought on Nicky Law and Kenny Miller with 25 minutes remaining as his side searched for an equaliser. The substitutions brought about more chances, but Falkirk keeper Danny Rogers was equal to every effort that came his way.
With the visitors' hopes fading, Rangers were offered the chance to salvage a point in the dying moments when Collum awarded a penalty after Blair Alston handled the ball in the area.
Waghorn stepped up for the kick but once again Rogers was the hero as his save in the right corner secured all three points to bring Falkirk within two points of the top two and condemn Rangers to just their second league defeat of the season.
The result means that chasing Hibernian can go level on points with the Glasgow side at the top of the division if they defeat Queen of the South later this afternoon.