Oshie opened the game's final segment with a backhander under the crossbar at the left post, and Shattenkirk ended it to start the third round with a low, hard slapper from the slot. In between, Ryan Miller stopped Vincent Lecavalier and Claude Giroux.
Miller came up with 31 saves in regulation for the Blues, whose 51st win tied them with the 1999-2000 Central Division champions for most in team history.
St. Louis also trails idle Boston by a single point in the race for the Presidents' Trophy.
Ray Emery stopped 28 shots for the Flyers, who have lost four of five but only two of those in regulation (1-2-2).
Philadelphia is currently a point behind the Rangers, who are playing Vancouver, for the second spot in the Metropolitan Division.
Miller came up huge early, stopping Michael Raffl from the left side to thwart a 2-on-0 break in the first 2 1/2 minutes. Giroux had a better angle and clearer shot but elected to pass.
Several close calls on either side did not result in a goal, with Emery halting 11 pucks and Miller making 10 stops in the game's opening 20 minutes.
The Blues rang two shots off the post in the middle frame and had the better of the chances, but 16 more shots (eight apiece) failed to find the net.
St. Louis effectively killed off a high-sticking minor to Vladimir Sobotka with 4:57 on the clock, then got a prime chance to win the game when Giroux was sent off for the same infraction with 1:24 remaining.
Miller slid from left to right to stop a one-timer by Nicklas Grossmann from the circle with 2:19 to go in overtime. He then dove forward to smother two in-close tries by Jake Voracek with 90 seconds on the clock.