The Tayside club came from behind after the break to overwhelm Allan Johnston's Ayrshire visitors with a late flurry of goals.
United are the form team in the country right now and McNamara expressed his satisfaction at progressing, with full-back Andrew Robertson, a summer signing from Queens Park, grabbing a double.
Robertson's opener was overhauled by goals from Darren Barr and Chris Johnston, but Stuart Armstrong drew the sides level again before Robertson grabbed his second of the night to make it 3-2.
Substitute Brian Graham's penalty and Gary Mackay-Steven's drive secured victory.
And McNamara said: "It was an exciting night. I think the game had everything. I thought it was a great cup tie. I would give credit to Kilmarnock for that second half, although they sat in a bit in the first.
"They got their tails up after the first goal and then scored a fantastic second goal through the young lad (Chris) Johnston.
"We had lost our way a wee bit at that point but I was delighted with the response.
"It was similar to the Partick game (which United won 4-1) in the way we just kept going and going. We got our reward for that.
"I am delighted to be in the next round. I thought it might have been one of those nights but fortunately it went for us in the end and I can enjoy my weekend now.
"I didn't bring Andy here for his goals but I am glad to have them.
"He was up in their box for the first one but that's what he is like in training - he wants to be involved in everything. He wants to score goals.
"I thought young Ryan Gauld was quiet tonight and he was maybe feeling the wee knock he got on Saturday. He was still involved in three of our goals, though."
Killie boss Allan Johnston admitted he was frustrated for his players on the night.
Johnston said: "I feel sorry for the players. They put in so much effort but it looks like they have been hammered. In reality, it wasn't really a hammering because we played a lot of good stuff. I think we can take a lot of heart from our performance.
"There were some positive parts which we can take in to the rest of the season as we had a very young side out there."