The Swifts thought they had taken the lead early on, but Danny Wallace's finish was ruled out for offside while, at the other end, Dunne produced a string of fine saves to deny Will Patching and Senan Devine.
Former Ross County man Henderson was only on the pitch a matter of minutes following an injury to defender Cameron Stewart, but he showed plenty of composure to control the ball before drilling a low shot out of the reach of Declan Dunne to give the away side the lead just before the break.
Kealan Dillon passed up a great opportunity to level for the hosts within minutes of the restart but his rising shot from 12 yards failed to trouble Coleraine goalkeeper Lewis Webb and they were made to pay shortly after as the Bannsiders doubled their advantage 10 minutes into the second period.
Doherty and Will Patching worked a short corner, allowing the former Derry City man to step forward and curl a delightful left foot shot into the top corner from the edge of the penalty area on his second debut for the club.
The home side were thrown a lifeline midway through the second half when Sean McAllister picked up possession midway inside the Coleraine half before firing low past Lewis Webb from fully 25 yards.
But hopes of a comeback were snuffed out by a third Coleraine goal with 12 minutes remaining.
Substitute Lewis McGregor delivered a low pass across the front of goal and there was the ever-reliable Shevlin to open his account for the new season to give the away side more breathing space.
Disappointed Dungannon Swifts boss Rodney McAree admitted there were plenty of positives to take from the defeat.
