Just as well too, as bottom of the table Gretna also took maximum points against Falkirk, to leave the gap unchanged at 12.
Motherwell - who dropped from third to fourth on goal difference - conceded early at Hearts last week and were caught cold again in Paisley, although full credit should go to the Buddies for a razor-sharp passing move which set up Mark Corcoran for a low left-foot finish.
Andy Dorman and Craig Dargo also threatened in a highly satisfying opening spell for the home side, but the feelgood factor quickly evaporated as Motherwell drew level with a stunning equaliser.
Steven McGarry created it with perfect control and a delicate pass and David Clarkson did the rest with an angled right-foot effort which curled into the far corner.
It was perhaps inevitable that the pace would eventually slacken and a more cagey period followed, before Saints moved through the gears again on the half-hour when Dorman's volley was cleanly held by keeper Graeme Smith.
But Smith had no chance seven minutes later as Saints went ahead for the second time.
Corcoran played in a deep corner kick and Maxwell met it at the back post, squeezing a downward header past a couple of defenders on the goal-line.
Shortly before half-time Corcoran's low free-kick almost deceived Smith as it flashed across the six-yard box.
Scotland manager-elect Mark McGhee would have still fancied his team's chances at that point, but a third home goal, early in the second period, dramatically altered the picture.
Again, Corcoran was instrumental with a neat pass to Maxwell, who disproved the theory about defenders being poor finishers, with a clinical drive past Smith.
Dorman nearly made it 4-1 on the hour mark only to see his shot deflected wide.
Six minutes later Motherwell's best move since Clarkson equaliser went unrewarded when a teasing cross from substitute Jamie Murphy found no takers inside the penalty area.
Top scorer Chris Porter was then denied by some top-class defending and a one-handed Smith save as Well battled in vain to get back into the contest.