Carlo Colaiacovo provided the other score for the Blues, who successfully followed up a 1-0 win over Chicago on Friday.
Chris Mason stopped 19 shots for his 10th win of the season and second in as many starts.
Lauri Korpikoski had the lone goal for the Rangers, who have dropped three games in a row and eight of their last nine.
Henrik Lundqvist allowed both goals on 33 shots and suffered his third straight defeat.
The Blues picked up the go-ahead score with 5:14 to play in the contest. Jay McClement won a draw and slipped the puck back to the left point. Barret Jackman lofted a shot from the blue line that got through traffic and skidded to Crombeen for an easy rebound conversion from outside the crease.
"It was a great job by Jackman. He found a hole and got it to the net," Crombeen said. "Luckily, I was in the right spot at the right time."
New York was credited with only four shots on goal in the third period and didn't manage a quality chance in the final 1:04 with Lundqvist on the bench for an extra skater.
"We had better legs than they did in the third," admitted Blues head coach Andy Murray.
St. Louis snapped a scoreless deadlock with 9:33 left in the second period, as Andy McDonald won a left-circle draw back to Colaiacovo for a blast which sailed over Lundqvist's shoulder.
Korpikoski knotted the contest with 3:13 remaining in the second, on a turnaround shot from the left circle.