The visitors were the first to show when John McGinn set up Kenny McLean but he missed his kick from the edge of the box and the chance went begging.
Brown took advantage of the let-off and duly gave his side a fourth-minute lead when he latched onto a cross from the left by Anthony Stokes to score from 15 yards.
St Mirren, led by caretaker-manager Gary Teale after Tommy Craig was sacked on Tuesday, were back on level terms in the 10th minute though.
McLean had a goal disallowed for offside after McGinn again played him in, but the Buddies were not perturbed and broke to score through Sean Kelly, who converted via the post after Marc McAusland had headed the ball back across goal.
But parity proved to be shortlived as Emilio Izaguirre picked out John Guidetti and he played in James Forrest, who scored from the corner of the six-yard box, also with the help of the upright.
Forrest cut in from the right before playing in Brown, who curled the ball around a crowd of players and inside the near post of St Mirren goalkeeper Marc Ridgers.
Ronny Deila’s side continued to pile on the pressure before the break, as Nir Bitton, Guidetti and Stefan Johansen all spurned further chances to score.
Stokes also caused problems after the restart, with the Republic of Ireland international seeing an effort chalked off for offside before curling a free-kick just wide of the target.
But the striker was not to be denied and he pounced midway through the second half, slotting home after latching onto a mistake by defender McAusland after he had lost possession.
Guidetti was forced off injured and his replacement Stefan Scepovic wasted a number of good chances to further punish a St Mirren side lacking in confidence.
The Serb twice shot wide when clean through before turning Izaguirre’s cross off the inside of the post, with the ball then bouncing to safety.
St Mirren rallied late on and almost pulled a goal back through McGinn, but he fired wide following good work from Jason Naismith as the Paisley side ended the weekend at the bottom of the Scottish Premiership table.
The victory leaves Celtic six points clear at the top after second-placed Inverness lost 4-0 at home to Partick on Saturday.