A spirited second half fightback from United proved not quite enough to rescue a point after a disappointing first period saw them two goals down at the interval.
Matching up with the home side's 4-4-2 formation, manager Peter Houston made only one change to the side that won so convincingly at Airdrie the previous week; with Neilson returning at right back, Flood pushed into midfield and Armstrong dropping onto the bench. Celtic made five changes to the side that recorded a comfortable cup win at Peterhead.
Though Celtic started brightly and dominated possession in the opening stages, it was the visitors who had the first sight of goal. Mackay-Steven's cross from the left was mis-hit by Robertson, but Kenneth - still forward after a corner - volleyed goalwards from an acute angle, forcing Forster into a sharp save.
United went on to force several set piece opportunities - but that was pretty much the visitors' only chance in a disappointing first half. Celtic took the lead on twelve minutes. After United lost possession, Stokes was able to put Hooper in behind and the striker expertly lofted the ball over the onrushing Pernis into the top corner. Just four minutes later and the lead was doubled - United were caught napping at a short corner, and Izaguirre's cross picked out Wanyama's late run with the midfielder bulletting an unstoppable header past Pernis from eight yards.
The remainder of the half saw Celtic content to play keep ball until the stroke of half time saw Daly shoot just off target from the edge of the box and Stokes glanced a header wide at the other end.
United replaced Flood with Armstrong at the interval but Celtic started the second period keen to finish the game as a contest - Mulgrew and Samaras both forcing good saves from Pernis.
The game took on a whole new complexion on fifty minutes as John Rankin brought the visitors right back into the game. The midfielder's stunning low strike from at least 35 yards rattled Foster's right hand post before nestling in the back of the net.
Suddenly United were pinning an increasingly nervy Celtic back and soon after the goal Foster did well to tip Daly's header over from a superb Dixon cross. Armstrong put a difficult chance wide and Russell was caught just offside in the six yard box as Celtic struggled.
Yet the home side remained dangerous on the counter and Samaras perhaps should have done better than head straight at Pernis when picked out by Matthews, and the Greek then blazed over from the edge of the box after being teed up by Brown.
Nonetheless, with the game now entering the closing stages, Celtic looked to have regained their composure and - try as they might - United just could not force that one clear-cut chance that might have capped an excellent second forty-five.
Teams:
Celtic: Forster, Izaguirre (Commons 78), Mulgrew, Rogne, Matthews, Brown, Ledley, Wanyama, Samaras, Stokes (Ki Sung-Yeung 65), Hooper.
Subs not used: Zaluska, Wilson, Cha Du-Ri, McCourt, Keatings.
Dundee Utd: Pernis, Dillon, Dixon (Douglas 64), Kenneth, Neilson, Flood (Armstrong 46), Robertson, Rankin, Mackay-Steven (Swanson 77), Daly, Russell.
Subs not used: Banks, Ryan, Armstrong, Dalla Valle, Dow.
Read the full report by Stuart McDonald