Clyde were 17 minutes away from causing a major shock at Ibrox before holders Rangers came from behind to progress to the quarter-finals of the CIS Cup.
Rangers debutant Federico Nieto scored twice in extra-time
Thomas Buffel slotted home an early opener for Rangers, but Craig Bryson bundled in a second-half equaliser.
Stephen O'Donnell put the First Division side ahead from the spot.
But Buffel pounced on a mistake to put the tie into extra-time, during which Federico Nieto scored twice and Marvin Andrews added another for Rangers.
Rangers manager Alex McLeish had promised to rest some players and give other fringe men a chance.
Yet summer signing Jose-Karl Pierre-Fanfan was still left warming the bench as 17-year-old Alan Lowing was given a debut.
Buffel, who was dropped for the Champions League win against Porto, opened the scoring with just five minutes on the clock.
Nacho Novo spotted the Belgian unmarked inside the box and all that was required was to slot past goalkeeper Paul Jarvie.
And Francis Jeffers was not too far away from his first goal for Rangers when he met a Bob Malcolm free-kick only to see his scuffed effort touched just wide by the goalkeeper.
Peter Lovenkrands was unfortunate not to add his name to the scoresheet with a far-post header that bounced wide.
But Ibrox was stunned into silence six minutes after the break, when former Rangers midfielder Tom Brighton squared the ball into the path of Bryson, who bundled the ball over the line.
Clyde had just three registered players in the summer and new manager Graham Roberts, the former Rangers defender, was forced to hold X-Factor style trials in order to build a team.
But Rangers boss Alex McLeish showed them respect by bringing on captain Barry Ferguson to replace Malcolm, and handed Nieto his debut in place of Jeffers, as he responded to the threat from the young Cumbernauld side.
A Lovenkrands effort curved towards goal only to be hacked off the line by Eddie Malone.
Clyde took a surprise lead when Lowing was guilty of a rash lunge on Robert Harris and O'Donnell converted from the spot after 73 minutes.
But Rangers were back on level terms a minute later when Jarvie allowed a long ball from Lowing to slip through his legs and Buffel pounced to fire home into the unguarded net.
Rangers edged in front eight minutes into extra-time after Lovenkrands chested the ball down for Nieto to bullet home.
The Argentine striker forced Jarvie into a flying save before Andrews effectively killed off the tie.
Nieto made the scoreline look more respectable when he nodded home a Lovenkrands cross.
Rangers manager Alex McLeish:
"I wanted the players to give me problems in terms of picking the team and I haven't been left with too many after that.
"It's annoying when you have to throw Barry Ferguson into the game, because that is not something I wanted to do.
"We must produce better than we did, despite the fact we made five changes.
"It was a great effort from Clyde and their performance and work-rate were unbelievable."
Clyde manager Graham Roberts:
"I thought we were going to win the game and that it would be history in the making.
"I'm a father figure to these lads, some of them are young enough to be my son, and I was very proud of every one of them. We gave Rangers a shock.
"When Barry Ferguson walked off the pitch at the end of the game, he turned around and said to me: 'You should have won'."
Rangers: Waterreus, Alan Lowing, Bernard, Andrews, Kyrgiakos, Novo (Namouchi 83), Malcolm (Ferguson 66), Murray, Lovenkrands, Buffel, Jeffers (Nieto 62).
Subs Not Used: Klos, Pierre-Fanfan.
Booked: Alan Lowing.
Goals: Buffel 5, 74, Nieto 98, Andrews 111, Nieto 113.
Clyde: Jarvie, McGregor, Harris (Brawley 75), McKeown, Malone, Bryson (Dick 90), O'Donnell, McHale, McGowan, Brighton (Arbuckle 68), Williams.
Subs Not Used: Miller, Cherrie.
Booked: McGowan, Malone, McGregor.
Goals: Bryson 51, O'Donnell 73 pen.
Ref: M McCurry