After a goalless first half, Joe Gallagher put the home side in front on 47 minutes, when he picked the ball up on the edge of box and fired a shot into the roof of the net
Eight minutes later Jamie Patterson headed in at the far post to double Edgehill's lead.
The win was sealed 10 minutes from time when Sean Exley scored a thunderbolt of a shot.
Edgehill boss Steve Clegg said: "Itis Itis worked hard but tired in the second half.
"Ryan Link and Lloyd Henderson shared our man of the match award."
Rovers boss Curtis Rose said: "New signing Connor Avison at centre-back stood out for us. Credit to Edgehill who were a lot stronger than us today."
West Pier also started 2022 in fine style with a 6-2 home win against fellow strugglers Scalby.
Zam Deans opened the scoring for the villagers with a 30-yard shot, but Dec Richardson netted twice to put Pier 2-1 in front at the break.
Harry Ward then scored the goal of the game for Pier, beating six players before finishing with aplomb
Ward then added his second, Richardson also bagging two more to bring his tally to four.
Deans netted his second with a free-kick for a Scalby consolation.
Pier's men of the match were Blake Drury and Dec Richardson.
Newlands claimed a 7-3 win at Beckett League outfit Snainton in the Scarborough FA District Cup.
Tristan Mustoe opened the scoring for the visitors, but Snainton fought back to go 2-1 ahead.
Mustoe was then fouled in the area and a penalty was awarded, Dan Freer slotting the ball home for Newlands.
After the interval Harry Holden netted a brace, the second a left-footed thunderbolt, to put Newlands 4-2 ahead.
Matthew Grffiths then scored a tap-in to make it 5-2, but Snainton refused to submit and scored a penalty to reduce the arrears.
Drew McCoubrey added a sixth after good work from Ben Luntley, and Freer scored a good back post volley to make it 7-3 at the final whistle.
The joint men of the match for the victors were Luntley and Brad Rowley
In the Scarborough FA Junior Cup, Goalsports suffered a 5-1 defeat at Kirkbymoorside Reserves.
The visitors paid the price for handing the Moorsiders several goals through errors.
Kieran Grzesiowski scored Goalsports' goal from the penalty spot.
Winger Sam Thompson was man of the match for Mark Plumpton's side.
Newlands Reserves also bowed out with a 6-1 defeat at Thornton Dale.
Kieran Wade scored for Newlands with Finley Davison-Ward the away team's man of the match.
Ryedale Reserves v West Pier Reserves and Edgehill Reserves v Newby were both postponed due to waterlogged pitches as was the Division Two game between Filey Town Reserves and Seamer Sports Reserves.