The Dons were on the back foot for much of the game, and Jamie Langfield made a crucial second half save at the feet of David Goodwillie.
"The defining moment of the game for them was Goodwillie's opportunity, and it's a great save from Langfield," McInnes told BBC Radio Scotland.
"We've took our glaring chance, and they've missed theirs."
Zola grabbed his second goal of the season with a close range finish after good link-up play between Peter Pawlett and Niall McGinn.
Jackie McNamara's side had the majority of the chances, but they couldn't find their way to goal.
McInnes said: "We started the game the better side but United finished the first half better - not through any great passage of play from them, just from us causing ourselves one or two problems from playing in the wrong areas.
"I think we started the second half excellently. We got a real grip of the game, and it culminated in a real bit of quality.
"The game was screaming out for a real bit of quality, someone cutting the defence open, and we managed to get that."
The Dons have now gone two months without conceding at home, and the Pittodrie gaffer said his back four have been the key to the Dons current top half position.
"We'll play better and not win," he said. "It was the type of game where possibly we could have buckled under not playing brilliantly and maybe made mistakes. We didn't make many mistakes even though we were not always great.
"We were up against a team who normally score goals, very rarely do they fail to score.
"We've chopped and changed with different personnel. We've six different defenders on the books who can come in and play. I though the back four today done their job well."
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