The champions responded well to losing an 11th-minute goal to Tannadice midfielder Stuart Armstrong with efforts from Efe Ambrose, Kris Commons and Joe Ledley.
However, at 2-1, the visitors were handed the chance to draw level when referee Bobby Madden surprisingly pointed to the spot after Celtic defender Adam Matthews tackled Gary Mackay-Steven, but Forster saved from United skipper Jon Daly.
Commons grabbed his second from the spot 10 minutes into the second half before Anthony Stokes notched a double to complete the rout, with Johnny Russell scoring late on for the Taysiders.
There were suggestions later that Madden had awarded the penalty for tugging, which was not lost on Lennon. He revealed on Friday that Celtic would be seeking "clarification" over Spanish referee Alberto Undiano Mallenco's performance during the 3-0 Champions League home defeat by Juventus on Tuesday night.
The Celtic boss laughed when he said: "I wish Bobby had been refereeing on Tuesday night. It was never a penalty. It is a great recovering tackle. I've seen it again. I think he (Bobby) said he tugged his shorts or shirt but he didn't, he recovered brilliantly and won the ball cleanly, there was no contact.
"We scored six goals and it could have been a lot more."
Jackie McNamara was "disappointed" after suffered his first defeat as United manager following wins against Rangers and Hearts.
The former Celtic defender said: "I was disappointed with the goals we lost and they obviously got a lift from us missing the penalty.
"I thought their keeper had a few good saves but the goals we lost were soft."