Gary Mackay-Steven, John Guidetti and Jason Denayer all scored in the first half at Celtic Park as the Hoops took advantage of Aberdeen’s earlier draw with Dundee at Dens Park.
Stuart Armstrong and Mackay-Steven were facing their former club for the first time since arriving at Celtic at the start of February, and both looked lively in this fourth clash between the teams in 14 days.
It was Mackay-Steven who opened the scoring on 16 minutes when Efe Ambrose played the ball inside from the right flank and the winger dispatched a low dipping shot which beat Radoslaw Cierzniak at full-stretch at the far post.
Three minutes before that Armstrong had tried his luck with a volley from 20 yards but sent the ball over the bar, and the Celtic midfielder later curled another effort from distance wide of goal.
In between those attempts Cierzniak had to be alert to deal with a Nir Bitton shot from 25 yards which bounced in front of the United goalkeeper.
Celtic doubled their advantage on 33 minutes when Mackay-Steven slipped the ball to Guidetti just outside the penalty area and the striker smacked home a shot which may have taken a slight nick off John Soutar.
Stefan Johansen hooked an effort over the bar and was later denied by Cierniak after Armstrong cut the ball back for the midfielder, with Mackay-Steven hitting the outside of the post with the rebound.
United midfielder Paul Paton was booked for a tackle on Mackay-Steven and from the resulting free-kick Johansen played the ball to Virgil van Dijk, whose shot broke off a United defender to Denayer (45) who netted with a cheeky back-heel.
United enjoyed more possession after the break and Craig Gordon comfortably saved a shot from Charlie Telfer before the Celtic keeper had to deal with a drive from John Rankin.
The visitors had another effort on 62 minutes and this time Gordon made an excellent save as he tipped a Ryan Dow shot over the bar.
At the other end Armstrong curled the ball over the target after Guidetti set him up as Celtic again began to stamp their authority on the match.
Kris Commons almost added a fourth when he latched on to a clipped pass from Scott Brown but Cierniak advanced to block at the feet of the Celtic substitute, and Soutar got back to stop the ball crossing the line.
The victory secured an aggregate 10-1 scoreline for the Hoops in the eventful mini-series which saw them draw 1-1 at Tannadice in the Scottish Cup, win 2-0 in the Scottish League Cup final at Hampden and chalk up a 4-0 replay win in midweek.