Alex Tettey, a Ghana-born Norway midfielder, signed from Rennes, scored for the Canaries as Rovers failed in their bid to reach the last 16 of the competition for the first time in seven years.
The League One side almost snatched an equaliser with two minutes remaining after full-back James Husband darted down the left and his low cross was stabbed goalwards by substitute David Syers, when better contact would have seen the ball nestle in the net rather than the grateful hands of goalkeeper Mark Bunn.
Saunders felt Doncaster did enough to get a positive result, but his side left Carrow Road with their heads held high after restricting the Barclays Premier League side to a 1-0 win.
"I don't like losing. If we had been beaten 3-0 I would probably have taken it better," said the Welshman. "It was there (for the taking) but we didn't have the killer instinct in front of goal. We must have created 10 chances, and have not scored. You have to be ruthless in front of goal.
"I am sick. I felt we could have won the game because they were not at full strength. It was an upset waiting to happen. We created the chances, but never took them."