Two of Watson's four previous goals came in the Europa League and another in a victory over Rangers and the right-back settled United down after a difficult start when he headed the opening goal in the 17th minute.
Dee, with Colin Nish serving a suspension, lacked bite up front but Steven Milne squandered an excellent first-half chance and they had a strong penalty claim denied when Brian McLean appeared to handle.
But they looked devoid of ideas before United put the game beyond them in the final 20 minutes through Jon Daly's penalty and a soft late goal from Willo Flood.
Both sets of fans in the 11,419 crowd paid tribute to former United captain and chairman Doug Smith, who died on Wednesday, with a minute's applause before kick-off.
United took quarter-of-an-hour to make an attack of note and they scored from their second corner moments later. Barry Douglas swung the ball out towards the near post and Watson peeled off his man to stoop and head home via a deflection.
The hosts created a great chance around the half-hour mark when Jim McAlister headed Gary Irvine's cross into the path of the unmarked Steven Milne but the forward volleyed over from 10 yards.
Dundee were being restricted to long-range efforts and the game went beyond them in the 71st minute when Willie Collum pointed to the spot.
Douglas saved from Daly after Stuart Armstrong had charged down Kevin McBride's clearance and cut the ball back, but Irvine had gone in late on the United midfielder. Daly had another chance to beat Rab Douglas and guided the penalty far enough away from the former Scotland goalkeeper's dive.
United made it 3-0 in the 88th minute when Flood took the ball from Armstrong 18 yards out and hit a weak effort that bobbled over the outstretched arm of Douglas and trundled home.