The Frenchman created United's first with a neat flick for Scott Fraser, who calmly slot into the far corner.
In the second half Andreu scored the second with a wonderfully measured chip from the edge of the box.
United lost Mark Durnan late on to a second booking and St Mirren were unfortunate not to convert one of several chances.
While most of the Buddies' first half opportunities were from long-range, in the second half both Lawrence Shankland and Tom Walsh had goal-bound chances cleared off the line.
Lewis Morgan was especially creative on the wings for the hosts - who are now without a win in their opening nine Championship matches.
Ray McKinnon's United are now four games unbeaten and the opening goal in Paisley was a great move, culminating in Tope Obadeyi finding Andreu at the edge of the box and the former Hamilton man skilfully setting up Fraser's assured finish.
Andreu's season-long loan from Norwich appears to be a brilliant piece of business with the 28-year-old on seven goals for the campaign already.
St Mirren goalkeeper Jamie Langfield had done well to block Simon Murray's effort after a poor backpass but the ball rolled to Andreu, who produced a deft chip of serious quality to find the back of the net from 18 yards to double United's lead.
Durnan's bookings were for shoving with Rocco Quinn and a foul on Shankland, but the Tangerines went down to 10 men too late for Saints to properly capitalise.
Dundee United manager Ray McKinnon: "It was a well deserved victory.
"The first goal was fantastic; we moved it from left to right side and back into the middle for Tony Andreu to play a lovely ball to Scott Fraser who finished well.
"The second one was a lovely to finish from Tony to dink the goalkeeper. He's a very, very talented footballer.
"There were four or five Premiership clubs (interested in Andreu) but we told him how we wanted to play and that he'd be hugely important to us. It's so important to have someone in your team that's got that little bit of quality."
St Mirren manager Jack Ross: "There were a lot more good things than negative things.
"There's a way I like to play and they are still adjusting to that. United were more clinical but we created more opportunities.
"If we produce a similar performance in weeks to come and we add a couple more things then we'll be okay."
Match ends, St. Mirren 0, Dundee United 2.
Second Half ends, St. Mirren 0, Dundee United 2.
Foul by David Clarkson (St. Mirren).
Blair Spittal (Dundee United) wins a free kick on the left wing.
Jason Naismith (St. Mirren) is shown the yellow card for a bad foul.
Cammy Bell (Dundee United) is shown the yellow card for a bad foul.
Attempt saved. David Clarkson (St. Mirren) right footed shot from the centre of the box is saved in the bottom right corner.
Second yellow card to Mark Durnan (Dundee United) for a bad foul.
Corner,  St. Mirren. Conceded by Paul Dixon.
Substitution, Dundee United. Coll Donaldson replaces Tony Andreu.
Mark Durnan (Dundee United) is shown the yellow card for a bad foul.
Rocco Quinn (St. Mirren) is shown the yellow card for a bad foul.
Jason Naismith (St. Mirren) wins a free kick in the defensive half.
Foul by Tony Andreu (Dundee United).
Attempt blocked. Tom Walsh (St. Mirren) right footed shot from the centre of the box is blocked.
John Sutton (St. Mirren) wins a free kick in the defensive half.
Foul by William Edjenguele (Dundee United).
Attempt saved. Lewis Morgan (St. Mirren) right footed shot from outside the box is saved in the top centre of the goal.
Attempt missed. David Clarkson (St. Mirren) right footed shot from the centre of the box is too high.
Foul by John Sutton (St. Mirren).
Frank van der Struijk (Dundee United) wins a free kick on the left wing.
Substitution, St. Mirren. David Clarkson replaces Gary Mackenzie.
Attempt blocked. Lawrence Shankland (St. Mirren) header from the right side of the box is blocked.
Corner,  St. Mirren. Conceded by Mark Durnan.
Goal!  St. Mirren 0, Dundee United 2. Tony Andreu (Dundee United) right footed shot from outside the box to the top right corner.
Corner,  St. Mirren. Conceded by Cammy Bell.
Attempt saved. Lawrence Shankland (St. Mirren) right footed shot from more than 40 yards on the left wing is saved in the top centre of the goal.
Foul by Charlie Telfer (Dundee United).
John Sutton (St. Mirren) wins a free kick in the defensive half.
Substitution, St. Mirren. Tom Walsh replaces Ryan Hardie.
Substitution, Dundee United. Blair Spittal replaces Tope Obadeyi.
Substitution, Dundee United. Simon Murray replaces Nick van der Velden.
Corner,  St. Mirren. Conceded by Paul Dixon.
Attempt blocked. Gary Irvine (St. Mirren) right footed shot from more than 40 yards on the right wing is blocked.
Substitution, St. Mirren. Lawrence Shankland replaces Stephen Mallan because of an injury.
John Sutton (St. Mirren) wins a free kick in the defensive half.
Foul by William Edjenguele (Dundee United).
Foul by Mark Durnan (Dundee United).
Stephen Mallan (St. Mirren) wins a free kick in the defensive half.
Foul by Willo Flood (Dundee United).
Tony Andreu was at the heart of Dundee United's win in Jack Ross' first game in charge of St Mirren.