With just five minutes on the clock, the home side got themselves in front.
The ball was crossed by Iestyn Hughes from the right and it made its way beyond the far post where Steven McCullough kept it alive by chipping it back across goal and there was Mulvenna with the header into the bottom right-hand corner for his third goal of the season.
The Bannsiders absorbed yet more pressure from the hosts, but were back on level terms on 22 minutes.
A long ball forward from Aidan Wilson found Shevlin on the edge of the area and he directed a low shot into the goal past a stranded Gareth Deane, who was caught in no-man's land.
Coleraine almost doubled their lead soon after when Joel Cooper received the ball down the left before crossing low to the centre of the 18-yard box. Thomson shot on goal, but his effort was blocked en-route to the goal.
Then, just before the half-time whistle went, Coleraine turned the game around.
Hughes - who has been so effective in attack for Bangor - dived in on Will Patching and the referee pointed to the spot. Up stepped Patching to slot home into the bottom right-hand corner give his side the lead.
The second period was a largely scrappy affair with few clear-cut chances, however Coleraine looked to have put the game to bed on 73 minutes.
The visitors won a corner that was expertly delivered from the left by Patching onto the head of Thomson and he guided home from six yards.
As all looked lost for the hosts, they grabbed a lifeline in the final minute of regular time and it came through a fantastic free-kick from Mulvenna.
The midfielder whipped it in from the edge of the area and it crashed off the post and into the net.
However, despite their best efforts, they couldn't find an equaliser and ended up empty-handed.