Ally McCoist identified David Templeton's introduction as the key moment in Rangers' Challenge Cup win over Hibernian.
A 2-1 extra-time victory for the Gers on Tuesday night booked a second-round date with Clyde, now managed by former Ibrox captain Barry Ferguson.
Hibs looked likelier winners when Danny Handling equalised a Lewis Macleod goal just before the hour, but Handling was then sent off for a foul on sub Templeton.
Nicky Law's 101st-minute strike then settled it Rangers' way, and McCoist said: "The change definitely helped us and the wee fella was absolutely terrific when he came on. David caused them all sorts of problems.
"We played reasonably well in the first half but Hibs certainly dominated the first 20 to 25 minutes of the second.
"We could have won with a chance right at the death through Kris Boyd but it needed extra-time and we kept going.
"It was two wonderful finishes from Lewis and Nicky and there was improvement in us but the most important thing was we won the tie.
"The main objective - and particularly in a cup match - was to get through and thankfully we did that.
"We can do better and we will do better. There's no doubt about that.
"We took our goals really well but we were sloppy for their equaliser. So there's room for improvement but I'm delighted with the result."
Hibs manager Alan Stubbs, who was taking charge for his first competitive game, said: "I have worked with the players for the last six weeks and this was where we were going to learn about them as individuals and as a team. They showed everything I had hoped.
"They showed a lot of courage to get on the ball and I felt they dictated the play at times. I thought they were cool and calm in possession. I thought we dominated possession.
"Coming to Ibrox and doing that, no matter what the Rangers team is at this time, takes a lot of guts.
"Look at the performance. Why should we fear anyone when we can play like that? We have come a long way in a short space of time and we will get better."