Goals from Sutton and Zaine Francis-Angol either side of a Gunnar Nielsen penalty save from Hoops forward Kris Commons gave the home side a two-goal lead before Anthony Stokes reduced the deficit just before the interval.
Georgios Samaras levelled 10 minutes after the break before Celtic substitute Leigh Griffiths, on for Beram Kayal, grabbed a third in the 84th minute, but as the game moved into injury time Sutton grabbed his second from a yard out to take the Steelmen to within three points of second-placed Aberdeen.
At the other end of the table, Aaron Taylor-Sinclair headed a late winner as Partick came from behind to climb out of second-bottom spot with a 2-1 victory over Kilmarnock at Rugby Park.
The Jags threw the race to avoid a relegation play-off wide open when Taylor-Sinclair nodded home from six yards following a cross by Kallum Higginbotham.
The goal offered redemption for the defender, whose error had allowed on-loan Brighton player Vitalijs Maksimenko to fire his first goal in British football in the 53rd minute.
The lead lasted just three minutes before Higginbotham got the goal his performance thoroughly merited as he netted from a free-kick.
Hibernian lasted just 12 seconds before conceding in their 2-0 loss to 10-man St Mirren as the Leith side were sucked even further towards relegation trouble.
The Easter Road men were in the race for a top-six slot just last month, but now find themselves in real danger of the drop after defeat in Paisley.
Kenny McLean fired the Buddies ahead with an instant opener before setting up Paul McGowan for the second after 14 minutes.
Hibs were given an opening when Saints skipper Jim Goodwin was shown a straight red card for a crude lunge on Kevin Thomson, but even with a man advantage they still could not find the net.
Hearts capped a momentous week off the pitch after second-half goals from Jamie Hamill and Dale Carrick wrapped up a 2-0 win which pushed Ross County into the relegation play-off place.
Hamill scored from the penalty spot in the 65th minute and Carrick wrapped up the win deep into injury time.
However, Hamill's strike from 12 yards was to be the midfielder's last involvement in the match after he needlessly collected a second booking moments later for his part in a touchline melee.
St Johnstone drew first blood against Scottish Cup final opponents Dundee United as a one-sided encounter at McDiarmid Park finished 2-0.
Tommy Wright's Saints side moved to within three points of United in the table thanks to goals in either half from Steven Anderson and Stevie May, who notched his 26th of the season.