A late double from Nicky Law and David Templeton sealed a 2-0 win and booked Rangers' ticket to the fifth round of the Scottish Cup at the expense of 10-man Falkirk.
The hosts played a man light for more than half-an-hour after David McCracken's red card but held firm until the 90th minute when Law broke the deadlock and Templeton then iced the cake.
Jon Daly had had a couple of early sighters while the returning Lee McCulloch headed over the top from a corner as Rangers pressed.
Falkirk were carrying a threat of their own through Rory Loy and Phil Roberts while Lee Wallace was seeing plenty of the ball down the left.
And it was down that flank that the marauding Rangers left-back carved out their first clear chance as he galloped clear but Michael McGovern was quickly off his line to block.
McCulloch shinned another great opening over the crossbar before the omnipresent Wallace blazed over the top from the angle.
The pressure finally told as McCracken was sent off for hauling down Nicky Clark, but McCulloch's penalty was tame and McGovern blocked with his legs.
Fraser Aird hit the bar from 18 yards as the battling Bairns' backline buckled, but they would not break - until Law's cool 90th minute finish from 12 yards.
In injury-time Lewis MacLeod teed up Templeton to sweep home the second past some tiring defenders to guarantee the 33-time winners' place in the last 16.