Gary Hooper's close-range strike after 23 minutes gave Neil Lennon's side the interval lead and second-half goals from Beram Kayal, Kris Commons and substitute Daryl Murphy, on for Scott Brown, sealed the win.
United substitute Johnny Russell grabbed a consolation goal for the Taysiders before Murphy struck deep in stoppage time.
It was a convincing performance from the home side, who could go top of the table on Wednesday when they travel to Inverness for their game in hand over the Ibrox men.
Before the game there was a minute's applause for former Celtic player Willie O'Neill, who died last week.
After the tribute, the home side started strongly and before two minutes had elapsed James Forrest, back in the side along with Anthony Stokes and Commons, had drawn a good save from goalkeeper Dusan Pernis with a powerful drive from 12 yards after cutting in from the right-hand side of the box.
The Taysiders were forced into some desperate defending as Celtic piled on early pressure and they were let off the hook in the seventh minute when Hoops defender Daniel Majstorovic headed a Commons corner over the bar from six yards.
Five minutes later the Scotland midfielder burrowed his way into the United box and his cross-cum-shot was re-directed by Stokes.
The ball clipped the outside of the far post before speeding wide and Peter Houston's side breathed again.
However, in the 23rd minute the visitors eventually succumbed to Celtic's pressure but it was United skipper Garry Kenneth whose carelessness gifted possession to Stokes wide on the left.
The Irishman hesitated at first but then set up Hooper with a pass across the six-yard box which allowed the former Scunthorpe striker to knock the ball into the net as Pernis scrambled to get back.
The goal changed the mood of the hitherto nervy home fans and, before United had a chance to compose themselves, Stokes sent a drive from distance just past Pernis' left-hand post.
United had been restricted to the occasional breakaway but, just before the break, striker David Goodwillie showed good strength inside the Celtic box to brush off the attentions of Kayal after racing onto a punt by Pernis.
However, his shot was blocked by backtracking Hoops skipper Brown and the chance of the equaliser was gone. United started the second half confidently and intent on getting back on level terms.
Morgaro Gomis had a decent opportunity when he was set up by Prince Buaben but the midfielder mis-kicked from 12 yards.
In the 51st minute Kenneth made some amends for his first-half error when he was in position to block Stokes' goalbound shot after Emilio Izaguirre's cut-back from the left had fallen to the Irishman inside the United box.
Then there was a penalty claim when Forrest's attempted cut-back appeared to strike the hand of Paul Dixon, but referee Iain Brines was having none of it.
However, any Celtic frustrations were swept away seconds later, in the 54th minute, by Kayal when he arrowed in a wonderful shot from 30 yards which gave Pernis no chance.
United responded by bringing on Jon Daly and Russell for Danny Swanson and Buaben but moments later, after more indecision in the United defence, Kayal had another effort on goal from the edge of the box, this time saved by Pernis.
On the hour mark Russell almost made his mark when he crashed a shot from 10 yards off the post after gliding past Brown and Charlie Mulgrew, before Celtic keeper Fraser Forster had to make a save from Gomis.
But United soon found themselves under pressure again as Celtic reasserted themselves in a bid to grab the third goal which would secure the points.
The home side continued to make good use of both flanks and one cross from right-back Mark Wilson found Hooper but his header was easily saved by Pernis.
In the 84th minute Commons drove in number three from 10 yards, after good work from substitute Shaun Maloney, who had replaced Hooper.
Russell grabbed a late consolation for the visitors before Murphy, on for Brown, restored the margin of victory with a fine solo goal and Celtic can look forward to the trip to the Highlands in midweek with some confidence.
Celtic 4-1 Dundee Utd: Match Report - view commentary, squad, and statictics of the game as it happened.