Kenny McLean opened the scoring with a well struck free-kick in the 32nd minute and Jon Robertson added the second with a superb volley 10 minutes later.
"It was a thoroughly professional performance, we knew there was an important job to be done today and that was to make sure we progressed to the next round," said Lennon.
"We did that and overall we were very, very good - the only thing I would say was that we weren't ruthless enough in front of goal.
"Once we got the goal it became a proper cup-tie as it forced Brechin out and we were able to pick them off.
"They made it difficult for us early on in the game, but I thought that some of our football was breathtaking at times."
Brechin boss Ray McKinnon said: "The first goal was really important, we had set up to counter-attack and a set-piece was our downfall.
"I have a really honest bunch of lads and they didn't want to lose face in the second half, they gave everything they had but St Mirren were worthy winners."
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