Four minutes later it was two to the good of the home team when Conway swerved a 25-yard shot which struck the palm of Dons keeper Jamie Langfield and found its way into the back of the net.
The visitors restored hope six minutes into the second half when substitute Josh Magennis headed home from close range after the Dundee defence failed to properly clear a long Ryan Jack ball before Scott Vernon played the ball across to make it 2-1.
But the Arabs rallied to essentially wrap up proceedings right on the hour-mark when David Goodwillie played a neat through ball for Danny Swanson who beat Langfield with a powerful left-footed strike into the corner of the net.
Dundee boss Peter Houston was delighted by the performance of his side who extended their unbeaten run in the league to four matches and are five points behind fourth-placed Kilmarnock with two matches in hand.
"I thought we were comfortable winners. I thought it was a very complete performance by Dundee United tonight," he said.
"I'm delighted with the players.
"I've always said if I can get Conway and Swanson on the park at the same time we'll cause team problems and tonight we did cause them problems.
"I'm delighted with the team effort tonight."