Calvin Zola turned home a Niall McGinn cross from close range after 53 minutes and though that proved the difference, Jamie Langfield made a brave save from David Goodwillie to preserve their lead.
McInnes said: "It was a hard-fought win against a decent United team. They've got good attacking options, and it's not often Dundee United fail to score.
"The teams had a bit of respect for each other - both sides have a threat, but I think for the most part that was cancelled out.
"If the game was screaming out for a bit of quality, that was provided in the build-up to the goal. Calvin thrives on service, and he got good service for the goal. He was where he should be at the right time, and made the difference in getting the three points.
"But the chance for Goodwillie was the defining moment. It was a great save from Jamie Langfield. He is as good as any Scottish goalkeeper in the SPFL. He's been brilliant to work with and is a great lad in the dressing room."
United boss Jackie McNamara lamented a lapse in concentration in the run-up to Zola's matchwinner.
"Frustration really sums it up. The players are a wee bit down and we're trying to get their heads up," he said.
"Sometimes it's wee things like a loss of concentration that cost you the match and that was the case today. We dominated the match and passed the ball well, and I don't think Aberdeen posed us many problems today.
"If anything we can be critical of our finishing, although Jamie Langfield had
a good save from David Goodwillie.
"I was pleased with certain aspects of our play, and I think If we play like that we'll win, or certainly draw, more games than we lose. We're creating chances, which is the most important thing."