Marsh ran for his first touchdown in the third quarter after completing his first touchdown pass to S.J. Green in the second quarter.
His last-gasp pass to Deslauriers set up the improbable comeback to the delight of the crowd of 22,456 at Molson Stadium, which saw Montreal overcome a 28-14 deficit.
Tyron Carrier returned a kick 90 yards to draw Montreal within 38-33 but the Alouettes failed to convert a two-point attempt.
Whyte had four field goals in the game.
The Alouettes (3-5) won for the first time in three games under general manager Jim Popp, who became Montreal's interim coach after he and owner Robert Wetenhall fired Dan Hawkins on Aug. 1.
Tim Brown returned a missed field goal 124 yards for a touchdown early in the third quarter and Travis Lulay threw a pair of touchdowns to Shawn Gore for B.C. (5-3), which had won two in a row.
Paul McCallum kicked five field goals for the Lions and Lulay threw his second touchdown pass to Gore, a 29-yard catch-and-run in the fourth quarter to put B.C. up 38-27.
McCallum, who kicked his fourth field goal at the end of the first half to put B.C. up 21-7, hit a 33-yarder 5:58 into the fourth. The veteran kicker also completed a 12-yard pass to Anton McKenzie for a first down on a fake punt in the second quarter.
Lulay threw a 10-yard touchdown pass to Gore in the first quarter.
Marsh, who made his pro debut in the Alouettes' 38-13 loss to Toronto on Aug. 8, took over from Josh Neiswander on Montreal's fourth offensive series.
Neiswander got his first career start in place of Anthony Calvillo, who was knocked out of Saturday's 24-21 loss at Saskatchewan with a concussion.
Intercepted twice in the loss to the Roughriders, Neiswander was picked off by Ryan Phillips four plays into the Alouettes' opening drive.
He completed one three-yard pass to Deslauriers in three attempt before he was replaced by Marsh.
Jerome Messam drew Montreal to 21-14 when he scored on a 16-yard run 38 seconds into the third after Carrier returned McCallum's second-half kickoff 65 yards.
Brown caught the ball deep in the end zone after a 53-yard field goal attempt by Montreal's Whyte went wide left and ran the length of the field to put B.C. up 28-14 at 3:21 of the third. Brown scored on a 97-yard kick return in the Lions' 26-22 win over Calgary on Saturday.
Arland Bruce, who passed Bryan Kelly to move into 10th place on the CFL's all-time receiving yards list, caught an 86-yard pass from Marsh 3:47 into the third. Marsh completed the drive with a six-yard run at 5:24 as Montreal closed to within a touchdown.
Duron Carter, the son of Pro Football Hall of Famer Chris Carter, made his pro debut and caught a 52-yard pass from Marsh on Montreal's second-quarter scoring drive.
Green celebrated his touchdown by tossing the ball into the end zone stands before going back to successfully implore the fan who caught it to return it for the rookie quarterback.
After McCallum opened the scoring with a 45-yard field goal off the upright, B.C. made it a 5-0 lead 8:07 into the game when Whyte punted the ball out of the end zone to concede a safety after a replacement long snapper Patrick Lavoie sailed a snap way over his head.
Martin Bedard, Montreal's regular long snapper, is on the nine-game disabled list with a shoulder injury. He was among eight Alouettes who were injured in Saturday's loss.