Celtic manager Neil Lennon hailed his team's attacking threat after they restored their five-point lead over Rangers.
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," he said. "In the first half they were sensational. 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 that they could have got another one, which they did, and really we should have picked Dundee United off a lot better on the counter-attack when they were throwing bodies forward.
"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."
Lennon played down a fresh injury to Scott Brown, who came off early in the second half after shaking off a groin problem to start, but Cha has a more serious issue following his return from South Korea duty.
"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.
"He felt it in the first day of training when he came back. He seemed to jar his tendon. It was pretty innocuous really because he came through the training session okay and only really felt it afterwards."
On Brown, Lennon said: "It was the other side. He felt a little tweak in his adductor area I think. We took him off as a precaution so hopefully he should be okay for the weekend."
Wilson evaded a second yellow card for a foul on Craig Conway just before the home side scored and, having been booked for his goal celebrations, he admitted Mike Tumilty could have sent him off.
"I was fortunate. It was a bad tackle, and I can only thank the referee for not giving me a second yellow," Wilson said. "I thought I could have won the ball. I didn't go in to hurt him or anything, but I was pretty lucky."
Lennon agreed with his player but United manager Peter Houston could not remember the incident and accepted they had lost to the better team.
"I don't think we turned up in the first half. I don't think we played anything near what we have been doing recently," Houston said. "For some reason we didn't come out the traps and if you give a good Celtic side a two-goal start you are always going to be up against it.
"When your better players can't pass the ball to each other, they are either nervous or Celtic put them under pressure. I felt Celtic put us under real pressure when we had possession.
"Rather than criticise my players, and yes they should have done better, I give credit to Celtic because they did the job properly."
Houston was pleased with the way his side fought back after Danny Swanson added some attacking spark to their midfield play but he added: "We didn't defend a Charlie Mulgrew free-kick properly.
"They attacked it and we were on the back foot and that's why we lost the third goal, and the game petered out after that."