The Bairns dominated much of the match without quite having the finishing touch their play deserved.
Twice Falkirk came close, with Neil McCann's shot clipping the bar and then a scrambled shot by Steve Lovell coming back off the post near the end.
Falkirk went close early on when McCann found space before unleashing a shot which clipped the top of the bar on its way over.
Falkirk wasted an even better chance minutes later when Darren Barr's cross eluded Madjid Bougherra. but McCann placed his right-foot shot just wide.
Just before half-time Rangers nearly took the lead when David Weir's net-bound header from a Steven Davis free-kick was headed clear by Lovell.
Rangers boss Walter Smith took Kris Boyd off at half-time and what an immediate impact Novo had.
From Sasa Papac's throw-in Novo spun and volleyed his shot from 30 yards which went over keeper Dani Mallo before dipping under the bar.
Falkirk looked stunned and Gerard Aafjes did well to keep out Kyle Lafferty as the striker forced his way through on goal.
Thomas Scobbie's runs down the flank were dangerous all afternoon and the wing-back saw his header drift narrowly wide following Jackie McNamara's cross.
Scott Arfield, who was tremendous in Falkirk's midfield, then saw a long-range effort go wide before the Bairns put the ball in the Rangers net, but Michael Higdon was clearly offside.
A header by Higdon was held by Neil Alexander before Rangers nearly increased their lead. A superb break by Kenny Miller led to his cross being deflected just over the bar by Novo.
Falkirk came so close to equalising against a tired looking Rangers team late on. Lovell's shot ran along the line with Bougherra stretching to touch it away before it hit the post.
It was a valiant effort by Falkirk but in the end it took one flash of inspiration from Novo to keep the Scottish Cup at Ibrox.