Another clean sheet means Rodgers' side have won every home league game bar a 2-2 draw with Aberdeen, having not conceded in any other Premiership match at Celtic Park this season, while United's defeat drops them to fourth place.
With Tuesday's game in Germany on Tuesday in mind, Daizen Maeda, Nicolas Kuhn, Alistair Johnston and Arne Engels started on the bench. It made little difference.
After initially showing stiff resistance, Jim Goodwin's side afforded Celtic captain McGregor too much space. The midfielder capitalised, holding off a couple of challenges before he reversed the ball just inside Jack Walton's post.
United struggled to threaten but Allan Campbell did have one chance in front of goal that he couldn't quite get on the end of.
McGregor then went close and Jeffrey Schlupp struck a post before Jota became increasingly influential.
His reward was an excellent second. Cutting in from the left the winger found a gap to curl the ball into the far corner beyond Walton.
The visitors stuck at it but were undone again late in the second half when Idah fired home a brilliant finish from the edge of the box to seal another three points.