Mallik Wilks continued his remarkable start to life at Rovers with his third goal in as many games as he helped his side into the second round of the Carabao Cup.
Wilks' first-half effort, coupled with a stunning Danny Andrew effort, was enough to seal victory at Glanford Park despite a late consolation from the home side.
Boss Grant McCann did not make huge changes to the side, with James Coppinger and Matty Blair came back into the starting XI in place of Alex Kiwomya and Tommy Rowe.
Rovers opened brightly, and had the first clear-cut chance after eight minutes when Ben Whiteman played Wilks through in a quick counter-attack - but the Leeds loanee's effort went over.
The home side's first effort came after 10 minutes when Lee Novak got a shot away - but a deflection took the power out of it and was easy for Marosi to collect.
Two minutes later he was forced into his first save of note, with Josh Morris trying his luck with a well-hit low drive.
But Rovers were looking dangerous on the counter-attack, with the front three of Marquis, Coppinger and Wilks looking dangerous.
Charlie Goode had to be alert to stop Marquis getting a shot away on a couple of occasions, with the No.9 firing over from a tight angle after winning the ball in the area.
Wilks again went close when Coppinger's chipped ball found Marquis, he fed Wilks who brought a good save out of Watson.
And Herbie Kane, on loan from Liverpool, almost built on his impressive debut with his first goal for the club as the game approached the half-hour mark.
Rovers attacked down the left and Wilks' cross found Coppinger. He did not have a shooting opportunity so laid the ball back for Kane, who cut inside onto his left foot and curled an effort just wide.
Kane had another chance moments later when a corner was not cleared properly, with the midfielder's effort blocked by a home defender.
Rovers took the lead after 36 minutes when the home side could not deal with a Rovers counter-attack. Coppinger's cross from the right was not dealt with, allowing Marquis to win the ball back and feed Wilks to drive home his third goal in as many games.
Rovers had the ball in the back of the net again after 43 minutes when Marquis headed home Andrew's cross, but the referee ruled the home side's Rory McArdle had been fouled in the build-up.
There was late drama with Andrew firing a shot towards the top-left corner as the two minutes of stoppage time drew to a close, forcing Watson into a great stop high to his right.
Rovers doubled their advantage 10 minutes into the second period with a goal which epitomised the attacking freedom McCann has given his full-backs.
Wilks had the ball in the centre of the pitch and fed Mason, whose deep cross was met by Andrew who acrobatically flicked the ball into the far corner.
The home side brought Stephen Humphrys off the bench in the search of a way back into the game, and an early foul on him allowed Morris an opportunity with a central free-kick - but his effort ended up with the home fans in the terrace behind the goal.
Rovers were playing with real confidence, with a Marquis header cleared from just in front of the line before Kane's drilled effort went wide.
Wilks was causing all sorts of problems and almost grabbed his second of the game with a low drive which went just wide of Watson's far post.
Alfie May, off the bench in place of Wilks, almost scored with his first touch when he got on the end of a cross from fellow substitute Paul Taylor - only to see his effort go just over.
The drama did not let up in the dying stages, with Rovers awarded a penalty after 87 minutes when Marquis ran at two defenders in the area.
Home keeper Watson was equal to it, though, getting down well to his right to rip Marquis' low spot-kick around a post.
And Marosi had to be alert moments later to get down well and tip a curling effort from Colclough to safety.
He was beaten two minutes into time added on when Humphrys curled a delightful effort into a far corner, leaving the Slovakian with no chance.
But Rovers saw out the game without any more danger, with Kane and Taylor both seeing efforts deflected behind.