Charlie Mulgrew scored the only goal of the Scottish Premier League game in the 15th minute and the Dons were looking comfortable against opponents who shipped seven at Rangers on Wednesday night.
But they had to cling on for the final half-hour after Davide Grassi picked up the club's seventh red card of the season for lashing out at Andy Webster, who was one of two United players to hit the woodwork late on.
With the big freeze continuing to wreck the Scottish football programme, the match had looked under threat, but it survived a 2pm pitch inspection.
That allowed Paul Marshall to make his Aberdeen debut after completing his loan move from Manchester City.
The Dons also recalled Jerel Ifil, Peter Pawlett and Fraser Fyvie, while United brought back Webster and Damien Casalinuovo.
Aberdeen would have been relieved today's fixture survived the cold snap following the postponement of their last two matches.
The snow finally abated as the match kicked off after a tractor had cleared the pitch an hour earlier.
But the skies were far from clear after a flare was let off in the end where Aberdeen's fans were housed.
They will have been content with their team's start and could have been celebrating in the 10th minute when Garry Kenneth's last-ditch tackle thwarted Fyvie in the box.
The visitors were ahead five minutes later when Marshall was given time on the edge of the area to feed Mulgrew, whose low left-foot shot arrowed beyond Nicky Weaver.
United could have quickly levelled when Webster beat Jamie Langfield to Craig Conway's free-kick but he was unable to control the ball before it went behind.
Webster was then shoved over the advertising hoardings at the other end by Darren Mackie, who was booked.
Dead-ball specialist Mulgrew fired in a free-kick which was parried behind by Weaver, who was playing the last match of his short-term deal at United.
An unmarked Morgaro Gomis fluffed a great chance to level 29 minutes in when he mishit Paul Dixon's cross at the back post.
Pawlett became the second player booked five minutes before the break when he clattered into Conway, moments after he was spread-eagled himself by Dixon.
Marshall had a chance of a debut goal a minute before the break but he was backpedalling trying to head in Richard Foster's cross before a Mulgrew free-kick forced an even better save from Weaver in first-half stoppage-time.
United emerged from their dressing room with far more conviction but they had another Mulgrew free-kick to defend six minutes in when Kenneth was booked for tripping Mackie.
This one dipped just too late as snow began to fall once more at Tannadice.
Pawlett flashed a shot across goal before being charged to the ground by Conway, who became the latest player carded.
United caretaker boss Peter Houston - possibly in his last match at the helm - then threw on Jon Daly and Jennison Myrie-Williams for Danny Cadamarteri and Sean Dillon.
The swirling snow forced the introduction of a yellow ball as the hour mark approached before Grassi's moment of madness at a United free-kick saw Aberdeen reduced to 10 men in the 63rd minute.
Webster was booked for his part in the scuffle.
United began to press and Andrew Considine almost deflected Daly's header past his own goalkeeper.
Visiting boss Mark McGhee withdrew Pawlett for Michael Paton before Marshall's deflected drive was well saved by Weaver.
Daly headed straight at Langfield and Houston made his final change as Danny Swanson replaced Conway.
The new man's corner was headed against the post by Webster and Daly steered a sublime first-time volley just wide.
Aberdeen sent on Chris Maguire for Marshall in the closing stages before Kenneth almost grabbed a dramatic equaliser when he headed Dixon's corner against the bar from just six yards.