Moussa Dembele netted the game’s only goal three minutes from time to set up an Aberdeen v Celtic final at Hampden next month.
Though Celtic dominated for long stages of the game, as Rangers stopper Matt Gilks made a number of good saves, the defeat still represents improvement on the 5-1 defeat suffered by Rangers at the home of their Glasgow neighbours last month.
Warburton feels the tight manner of the result illustrates the continued improvement of the Rangers players since that fateful day.
He told BT Sport: “The gap is a lot narrower. Our work in the last few weeks is hopefully there for all to see.
“We were a little bit disappointed with the play in the second half, we went long too often. But that’s the time of thing that happens in a game with a lot of emotion.
“It was an open second half, chances at both ends and unfortunately it went the wrong way. There were a lot of positives to take out of today.”