A fine second-half performance from Alan Archibald's visiting side put Motherwell under pressure and meant Partick could consider themselves unlucky to leave Fir Park with no reward.
Steven Lawless never made a league appearance for Motherwell before leaving in 2012, but on his return to the club he provided an early scare when his header back across goal was fumbled by Lee Hollis.
Home manager Stuart McCall left summer signings Paul Lawson and Iain Vigurs on the bench. The replacement for the latter, Zaine Francis-Angol, had the first opening for the home side but after working his way into the box he screwed his shot well wide of goal.
All was forgiven in the 20th minute when Motherwell opened the scoring. Anier laid the ball in from the left and Francis-Angol's touch and turn set up Sutton who buried the ball low and hard into the net from eight yards for the first goal of his second spell with the Steelmen.
For the first time this season Partick found themselves behind and Stephen McManus should have done better from a Stevie Hammell free-kick as the home side pushed for an equaliser. Unmarked, the former Celtic and Scotland defender headed weakly into Scott Fox's hands.
The goalkeeper came to Partick's rescue after 32 minutes when James McFadden's cross was cleared only as far as Anier and his low volley was destined for the corner until Fox leapt to his right to push the ball away.
The Partick defence looked a little ragged and another McFadden cross caused problems until Aaron Muirhead booted the ball away from inside the six-yard box.
With the Maryhill side holding out for the interval, Anier had another double attempt at goal. His first effort was blocked by Conrad Balatoni and he slammed the rebound low but straight towards Fox.
Early in the second half Sutton laid the ball off for Anier who pulled a shot wide from 20 yards, before McFadden headed Simon Ramsden's cross over the bar.
Former Aberdeen midfielder Isaac Osbourne produced Partick's first real strike on goal after 50 minutes but it was far too straight to pose any problem for Hollis.
McFadden then had two attempts. A terrific volley from Sutton's cross followed by a run and shot from 25 yards had the home crowd rising in anticipation of a goal.
With former Motherwell player Kallum Higginbotham coming on for Gary Fraser, Partick began producing goalmouth chances in front of their huge travelling support.
However, Fox had to keep the visiting side in the game with another superb save after 66 minutes. Anier unselfishly played the ball in front of Keith Lasley, and his shot from 12 yards was kept out by the goalkeeper's outstretched leg.
That gave Partick hope for the final 20 minutes, especially when Anier was withdrawn much to the disapproval of the home crowd.
When Partick brought on another former Motherwell player, Ross Forbes, the script looked to be set for a deserved equaliser.
One block on the line from McManus caused roars from the Partick supporters for a penalty.
Higginbotham then cut inside only to watch his thumping drive blocked by Hammell as Motherwell desperately held on for all three points.