Stephen O'Donnell opened the scoring from outside the box, Conrad Balatoni added a second before Mark Shankland pulled a goal back and later struck a post.
Archibald said: "It was a bit nervy in the last five minutes but I thought we had been comfortable at two-nil. That's what happens.
"We scored two smashing goals but they then played a long ball up the park and got a goal from it.
"That was a bit disappointing. But apart from that I was happy with the result. We're through to the next round and that is all that matters.
"It was heart in mouth a bit when they hit the post and I thought Ayr were very good.
"People might expect us to go out and run over the top of teams. I think that's disrespectful."
His opposite number at Ayr, Mark Roberts, insisted his side had done well enough to force extra time.
He added: "I don't like to use the old line about glorious failure but we did not look like we were two divisions apart.
"We could have taken it into extra time and Thistle were on the ropes.
"We had a couple of great chances and we just didn't get that bit of luck."