It was a comfortable night for the new management duo of Barnes and Jason McAteer after three goals in eight first-half minutes ended the contest before the interval.
Grimsby had started brighter and Rovers stopper Luke Daniels pulled off the first of several good saves from a point-blank Nick Hegarty header.
Rovers found their rhythm just after the half hour when Paul McLaren bagged the opener at the second attempt.
An Aaron Cresswell cross from the left on 32 minutes found McLaren unmarked to volley a thunderous effort from eight yards that looked a certain goal.
Grimsby's Irish stopper Nick Colgan miraculously reacted to push it up in the air but the Rovers man nodded in from inches into the empty net.
If going one down was unlucky for Mike Newell's Mariners, losing a second was harsh and they did within five minutes after referee Anthony Taylor adjudged left-back Joe Widdowson to have brought down Craig Curran inside the box.
Ian Thomas-Moore stepped up to slot home a low right footer into the bottom corner past Colgan's outstretched right arm.
After being in the driving seat, Grimsby had fallen apart and Rovers had a third five minutes before the break.
Centre-back Ian Goodison - back from suspension - was somehow allowed space on the left to cross and Curran was the latest unmarked man to profit inside Grimsby's box.
Grimsby could have had a way back into the tie on 50 minutes when a searching long ball found Adam Proudlock in space and through on goal but his effort was pushed away by the onrushing Daniels.
Grimsby substitute Danny North came on and injected life into the Mariners front line, forcing Rovers man of the match Daniels to tip over a header and then get down to stop another.
But a fourth goal went the way of home substitute Gareth Edds when Chris Shuker's out swinging corner fell his way 15 yards out and he arrowed a header in of the woodwork in the top corner.