Dods had fouled Barry Robson inside the box to give the former Tangerines midfielder the chance in the 71st minute to send Nicky Weaver the wrong way from the spot.
However, with seven minutes remaining, Jon Daly, on for Damian Casalinuovo to play his first competitive game since April following a knee injury, headed the equaliser from Danny Swanson's corner.
In the dying seconds, from another Swanson corner, Dods left the visiting fans stunned when he bulleted a header past Lukasz Zaluska.
Mowbray admitted the tinkering with his defensive tactics to counter-act the height and strength of United had failed but insisted his side should have still left Tayside with three points.
"I felt there was a huge gulf in quality and class between the two teams and yet they scored from two set-plays," he said.
"Whether they can continue to score from set-plays, I don't know.
"I don't think we got what we deserved from the game but if you don't defend your set-plays properly, then goals are what the game is about.
"But generally, for about 85 minutes, I think it was one-way traffic.
"We totally dominated the football match. The first half wasn't a problem for us. I never felt they could get near our goal.
"I didn't think they could hurt us other than set plays and ultimately they did, which of course, is sore.
"But they didn't create any other chances."
Mowbray added: "They have some huge, physical boys which is why we changed things today.
"Instead of man-marking, we went to part-zonal marking.
"We didn't have the physicality to mark their six foot four, six foot five players.
"So you try and out your strongest headers in the best part and most dangerous part of the pitch and in theory it works but in practice it didn't."
Celtic passed up on the chance to leapfrog Rangers to the top of the table but Mowbray claimed not to be too concerned.
He said: "I judge my team on the performance level and whilst we didn't get the result, it shows me that for the bigger picture, we should add a little bit of quality here and there this team can go on and be successful for a very long time."
Asked about Mowbray's assertion about the "gulf in class between the two teams," Dundee United manager Craig Levein joked: "We weren't that much better than them.
"That would be doing Celtic an injustice, really."
Levein continued: "He is entitled to his view of the game the same as I am and I didn't think we played very well but how many saves did Nicky Weaver make in the first half?
"I've played against Celtic in the past where the keeper has had to make wonder saves and we have still been 3-0 down at half time.
"Then, I would have said there was a gulf in class - but not today."
Levein, who refused to talk about the constant speculation linking him with the vacant Scotland post, was especially pleased for Daly.
"If there is such a thing as a footballing God then he is looking down fondly on Jon Daly today," the United boss said.
"He certainly hasn't been looking down very fondly on him in the past.
"The only person who will be happier than me is Jon.
"He has been out for seven or eight months and played 30 minutes of football in that time.
"I think he was surprised that he was on the bench - and I had him on the bench because we needed cover in central defence.
"He is away home absolutely thrilled and so he should be."
Dods was relieved that his mistake for the penalty had not proved decisive.
"It was disappointing to give away the penalty," he said.
"I played against Barry Robson for years and I know all about his step overs but he done me this time.
"But the win over Celtic has been a long time coming and I'm delighted."