Despite taking an early lead through Lewis Macleod at Ibrox, the Light Blues toiled against Alan Stubbs' side as they levelled through Danny Handling and passed up a number of decent scoring chances.
The introduction of Templeton and a switch to a 4-4-2 formation gave Rangers a timely boost and paved the way to victory for McCoist's side.
And the Ibrox boss was pleased to see his tactical switches pay dividends against the green half of the Capital as Templeton shone after stepping off the bench.
McCoist said: "We played reasonably well in the first half but Hibs certainly dominated the first 20 to 25 minutes of the second.
"The change definitely helped us and the wee fella was absolutely terrific when he came on. David caused them all sorts of problems.
"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."