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.
While Rovers created the first chance of the second half, this time it was Larne striking an early goal.
After a corner was cleared only to the edge of the box, Chris Gallagher, back after missing the Molde match through suspension, fired home despite Leon Pohls getting a hand to his effort.
With Rovers' vocal supporters quietened for the first time in the game, Tiernan Lynch's side will have had designs on clawing their way back into the game but instead they gifted their opponents a fourth just six minutes later.
Joe Thomson's pass in midfield was careless and Burke needed no second invitation to finally get the goal his performance deserved, picking up the loose ball and firing clinically past Ferguson.
The former Preston North End attacker, who won three caps for the Republic of Ireland during 2018, was replaced to a rousing ovation soon after.
The intensity in the game dropped across the final 20 minutes, although Ferguson still had to make saves from Darragh Nugent and Trevor Clarke to keep the deficit at three, while Andy Ryan had a chance for a consolation in injury time.
With four points from two games, Rovers have given themselves a real chance of progression in this new format of the competition, even if their focus now returns to a title race that sees them two points behind leaders Shelbourne with two games to play.