Goals from Sutton and Zaine Francis-Angol had put Stuart McCall's side 2-0 up, with Gunnar Nielsen saving a Kris Commons penalty before Francis-Angol doubled their lead.
Anthony Stokes then pulled a goal back in first-half stoppage time and the champions looked set to pull off a stunning victory when goals from Georgios Samaras and Leigh Griffiths put them 3-2 up with two minutes remaining.
Steven Anderson and the prolific Stevie May were on target as Tommy Wright's side secured a victory that leaves them just three points behind United.
There was controversy at Rugby Park as Partick Thistle secured a 2-1 victory over Kilmarnock to move level on points with the hosts in the battle to avoid a relegation play-off.
The home side were convinced that the ball had crossed the line before goalkeeper Paul Gallacher cleared, but referee Brian Colvin waved play on and Aaron Taylor-Sinclair rubbed salt in the wounds by scoring the winner five minutes from time.
Victory for Alan Archibald's side moved them above Ross County, who are now second from bottom after they were beaten 2-0 at relegated Hearts.
Kenny McLean scored after less than 15 seconds and Paul McGowan was also on target as St Mirren secured a 2-0 home victory over Hibernian, despite playing with 10 men for an hour following Jim Goodwin's dismissal.
