Goals from Brian McLean and Gary Mackay-Steven were enough to secure McNamara's side a semi-final spot, and the former Celtic star praised his side's performance.
"It was a typical cup tie and derby, not a lot of quality or passing. But credit to our lads, and Dundee as well, they put a lot of pressure on us and put extra men up front. I thought we coped with it really well," he said.
"They were on top and we scored a fantastic goal, a great run from Stuart Armstrong and a great pass to Gary Mackay-Steven.
"Dundee had been on top and it was frustrating because they were getting the second balls. But we battled away and got there in the end."
Dundee boss John Brown lashed out after the Dens Park tie, blaming assistant referee Andrew McWilliam for failing to flag up a throw-in moments before Brian McLean netted United's first goal.
McNamara, however, did not feel the incident was crucial to the scoreline.
"I saw them appealing but I didn't actually see it was out. I think things went for both teams, some decisions."