Larne, the first Irish Premiership team to make it to this phase of a major Uefa competition, had not looked overawed in their competition debut against Molde last month, keeping the Norwegians at bay until the 51st minute before ultimately losing 3-0.
Here, though, Stephen Bradley's four-in-a-row league champions looked a cut above from the off.
Indeed, the travelling Rovers fans were given cause to celebrate after only two minutes with Burke's effort after a driving run diverted back into the goalmouth where Josh Honohan tucked home.
With Burke proving particularly influential, Rovers found space in the box again soon after, although this time top-scorer Johnny Kenny could find only the side-netting.
While Benji Magee delivered a teasing ball that could have produced an equaliser, it was no surprise when Rovers doubled their lead after 24 minutes.
Burke was again the creator, drifting in behind when Larne failed to fully clear from a corner and then delivering a ball that left Kenny with the simplest of tasks to nod home into an empty net from all of a yard.
Having provided the first two assists, Burke thought he had turned goalscorer for Rovers' third, glancing the ball past keeper Rohan Ferguson via a deflection off Tomas Cosgrove who had been tussling with his man at the back post.
Initially chalked off for a foul, the goal was eventually given after a VAR review but would ultimately go down as an own goal by the Larne skipper.