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Lee had just seen a 25-yard volley by David Livermore give Millwall a 3-2 victory in the 90th minute but their opponents from League Two threatened an upset when they managed to claw back a two-goal deficit in the space of as many minutes.
First Simon Brown pounced in the 67th minute after Andy Marshall superb stop from Gus Uhlenbeek's long-range effort then just over a minute later Richard Baker equalised for Mansfield.
"A win is a win and it's nice to progress," said Lee after the match. "At 2-0 you never think it's game over because that's the wrong attitude to take and Mansfield really came back to bite us.
"We had four of our regular players missing for one reason or another but that's no excuse for what happened because we had too many who were below par out there on the park.
"And we know if we perform like that in the Championship then we're not going to be winning any games.
"But getting to the last 16 of the competition was a great incentive for us and if we get a big team in the next round then all well and good."
The Lions had taken the lead through Ben May, who headed home from close range in the first half and extended it further on the hour mark when Paul Robinson met Don Hutchison's corner to score. But it was Livermore's wonder-striker that secured the club's place in the Fourth Round of the Carling Cup for the first time in ten years.