The Gers got the better of the Scottish League One side at Ibrox in Glasgow in the fourth round of the prestigious competition.
Martyn Waghorn was the star of the show for Rangers, as he scored a hat-trick to help his team register a comfortable 5-1 win.
"It was a tough challenge, so we had to prepare well and perform well and we did that,” Warburton told BBC Radio Scotland. "We had players all over the park looking to score and get into scoring positions. We have options and I am delighted with the squad we have."
Harry Forrester made his debut for Rangers against Cowdenbeath, coming on as a substitute in the 62nd minute.
The 25-year-old attacking midfielder, who joined the Glasgow giants on a six-month loan deal from Doncaster Rovers earlier this month, has expressed his delight at turning out for the Gers on Twitter.
Delighted to make my debut, the reception was class..Thank you! Next round, cobwebs blown away for me, waggy hat-trick! Good day all round!— Harry Forrester (@harry_forrester) January 10, 2016
Over the course of the 90 minutes at Ibrox, hosts Rangers had as much as 69% of the possession, took 30 shots of which 14 were on target, and earned 12 corners.
Visitors Cowdenbeath had just 31% of the possession, took five shots of which one was on target, and earned two corners.