Cliftonville were first to threaten when Rory Hale forced Lewis Webb into a save inside the opening minute, but Coleraine responded to open the scoring.
Shevlin began the move, heading a long punt forward towards Jay Henderson and he then spotted the run of Joe Thomson, who was breaking into the penalty area.
The former Glentoran man showed great awareness to square the ball for Shevlin, who had the simple task of prodding home from six yards.
Ben Doherty and captain Joel Cooper went closing to doubling the home side's advantage in an entertaining opening, but they didn't have things all their own way.
Brian Healy crashed a low shot off the woodwork from the edge of the penalty area while Webb produced a number of fine saves to deny Harry Wilson and Rhyss Campbell.
Cooper had a great chance to double Coleraine's lead minutes after the restart when through one on one with Morrison, but the Cliftonville keeper did well to block the shot.
At the other end, Dean McMaster came within a whisker of equalising when his volley - after his initial free-kick had been blocked by the Coleraine defensive wall - flew narrowly wide of the post.
McMaster came closer still 10 minutes later when his clever free-kick had Lewis Webb scrambling across his goal line to gather at the second attempt as everyone expected a cross into the box.
Morrison proved his worth midway through the second half when he dived full length to his right to palm away a Cooper effort, before saving from Wylie with minutes remaining.
The visitors struck the woodwork for a second time in the game in the final minute of normal time when Shaun Leppard's towering header glanced off the top of the crossbar with the goalkeeper beaten.