Celtic remain at the summit of the SPL after a gutsy win at Tannadice.
Vennegoor of Hesselink opened the scoring at Tannadice
Dundee United captain Barry Robson hit the crossbar with a thunderbolt, while Grzegorz Szamotulski made a great stop from Jan Vennegoor of Hesselink.
But the Celtic striker was not to be denied and powered home a header from a Paul Hartley corner after the interval.
Stephen McManus knocked in a second goal with his chest soon after as United failed to deal with another wicked delivery from Hartley.
Defender Bobo Balde was handed his first start in a year as Celtic looked to overcome a sticky patch of results.
And his presence helped the visitors keep their first clean sheet away from home in almost 15 months.
Rangers had taken top spot for 45 minutes, having kicked off earlier, but Celtic are two points clear again, although their rivals have two games in hand.
The game started at a frantic pace but clear-cut chances were few and far between in the opening stages.
Celtic midfielder Jiri Jarosik should have done far better in the 22nd minute when, after latching on to a knock-down from Vennegoor of Hesselink, he drove the ball wide of the target from a good position in the penalty area.
Tannadice skipper Robson whipped a corner straight on to the head of Lee Wilkie but the defender turned the ball wide, with Celtic keeper Mark Brown rooted to his line.
Visiting captain McManus then rattled the United crossbar with a shot, but the referee Eddie Smith had already blown his whistle for an offside.
In the 38th minute Robson took aim from all of 35 yards out and his powerful drive left Brown helpless as it rebounded off the crossbar.
And an absorbing first half ended with Szamotulski producing an outstanding save to tip a towering header from Vennegoor of Hesselink up into the air and once again off the woodwork.
United looked the stronger side after the break but Celtic's Aiden McGeady threatened with a surging run and shot on 65 minutes that flew a yard or so wide of the target.
Soon after, Celtic were in front when Vennegoor of Hesselink rose highest to nod a Hartley corner home.
The visitors made the best of another set-piece a few minutes later and McManus arrived at the back post to bundle in a flick-on from Massimo Donati with his chest.
Celtic started to open up with great effect and a blistering run from McGeady set up Scott Brown for a shot that cannoned off the post.
The home side battled gamely in the closing stages but could not find a way back into the match.
Dundee Utd: Szamotulski, Dillon, Wilkie, Dods, Flood, David Robertson (Conway 58), Buaben, Gomis, Robson, Hunt, Jordan Robertson. Subs Not Used: McLean, Kerr, Cameron, Kenneth, Duff, Don.
Celtic: Mark Brown, Caldwell, Balde, McManus, Naylor, McGeady, Scott Brown, Hartley, Jarosik (Donati 72), McDonald, Vennegoor of Hesselink. Subs Not Used: McGovern, Riordan, Sno, Killen, O'Dea, Caddis.
Goals: Vennegoor of Hesselink 68, McManus 74.
Ref: E Smith