Super Gary Hooper fired home from close range after 23 minutes following a mistake by United defender Garry Kenneth.
Beram Kayal curled in a stunning second nine minutes after the break.
Kris Commons drove another home after 85 minutes before Johnny Russell’s late close-range reply for United and Daryl Murphy’s stoppage-time solo effort.
And Neil Lennon’s side can now leapfrog the reigning champions should they win their game in hand on Wednesday, when they travel to Inverness to face Caley Thistle.
Following the goalless draw with Rangers, the Celtic manager had restored Commons, Anthony Stokes and James Forrest to the starting line-up, replacing Georgios Samaras, Ki Sung-Yueng and the injured Joe Ledley.
Manager Neil Lennon was full of praise for Scott Brown and Beram Kayal after Celtic overpowered Dundee United.
“I thought the two boys in central midfield were fantastic,” a delighted Lennon told BBC Radio Scotland.
“They were absolutely immense. Kayal and Brown dovetailed superbly. That’s as good a midfield as we’ve had here for a long time.”
Kayal’s thunderous shot to make it 2-0 was the pick of the bunch and capped a wonderful display from the Israeli international.
“It was a wonder goal and thoroughly deserved for his performance overall,” enthused Lennon. “He plays with great intensity, application and quality.
“I played with great players – Paul Lambert, Stiliyan Petrov – I was very lucky in that respect. But these guys could potentially be as good.
“And we’ve got Ki Sung-Yueng to come on and Joe Ledley, who has been out, and Efrain Juarez as well.
“There’s a good blend there. They have real power and they can play. What we’ve lacked is players who can get the ball back for us and they do it very well.”
Celtic now face three consecutive away matches – at Inverness, Kilmarnock and Hearts – before their SPL campaign concludes with a home game against Motherwell.
A run of four wins would secure the title and Lennon took great heart from the manner of victory against United.
“Rangers are a strong, formidable team,” he said. “It was a bit of a surprise to see them win 5-0 at Fir Park, but we responded very well.
“I thought we were superb and it could have been a lot more.
“We’ve got good pace in the team, good tempo and we had a bit of aggression about us today.
“We played some great stuff and cut them open at will. Really, we were sloppy in the final third at times.
“We should have been a bit more clinical. But, if you’d given me 4-1 before the game, I’d have bitten your hand off.”
Lennon had a special word of praise for Murphy, who scored his first goal from open play for the club with a great solo effort in injury time, having struggled to command a regular place in the team.
“I was delighted for him,” added Lennon. “He’s never complained once. He trains really well and he did a great job for us at St Johnstone.
“You could see the reaction from the players and what it meant to them.”