Anthony Stokes and Mark Wilson benefited from deflections to put Celtic in command before the break and defender Daniel Majstorovic dived to head home a 78th-minute clincher after David Goodwillie had got United back in the game.
Celtic could have added more goals to their tally with Gary Hooper hitting the post, seeing a late goal ruled out for offside and completely losing his footing with only Dusan Pernis to beat.
Lennon has been hit by some bad news with full-back Cha Du-ri set for a lengthy lay-off with an ankle injury but he was delighted with the form of Hooper and Stokes as his team secured their fifth consecutive SPL win.
“I thought the front two set the tone. In the first half they were sensational,” Lennon said. “We could easily have gone in with a more comfortable lead.
“At 2-0 you are always concerned that Dundee United could get back in the game, and they did.
“But there was a real belief about the team.
“We got a bit of luck with the first two goals but I thought we earned it. We missed easier chances and should have been more clinical at times, but the fact we created quality chances really pleased me.
“I’m really pleased with the threat we have and we defended solidly too.”
Korean star Cha did pick up an injury to dampen Lennon’s delight.
“Cha took a bad one. We are sending him to see another specialist to get another opinion,” Lennon said. “We are hoping that he will not require surgery, but even if he doesn’t require surgery he will be out for a significant period of time.”
On Saturday, Rangers boss Walter Smith praised Kyle Bartley and David Healy after the January signings made impressive debuts in Rangers’ 6-0 romp over Motherwell.
Arsenal defender Bartley, who was on loan at Sheffield United before sealing a similar move to the Scottish champions, made his first appearance when he started in the holding midfield role usually occupied by the injured Lee McCulloch.
Healy also had a maiden bow to remember when he completed the rout at Ibrox following Steven Naismith’s opener, a hat-trick from Nikica Jelavic and an own-goal by Shaun Hutchinson.
Rangers will learn this coming week whether McCulloch will play again this season, when he goes under the knife to cure a knee problem, and Smith was asked whether Bartley could be a potential replacement.
He said: “The lad is a central defender, he is not a midfield player but he has played there before. We were keen to see whether he could fit into that role and I think he showed he can move forward quite comfortably into that kind of role.
“I thought Healy played very well and he could have had another goal if it wasn’t for an excellent save from the goalkeeper. The pleasing thing was managing to get the players we brought in in January at least integrated into our team and starting to play.”
Smith confirmed Naismith – who was withdrawn just before the break – should be fit for Thursday’s Europa League tie against Sporting Lisbon.