Rollins struck on the stroke of half-time to break the deadlock in a cagey derby affair at The Jakemans Stadium.
Brad Abbott subsequently scored his first goal in amber and black with 17 minutes to play as the Pilgrims wrapped up a routine win over their neighbouring Blues.
Paul Walker and Lee Bennett had been handed their first starts for the Pilgrims ahead of kick-off, with Brad McGowan ruled out for the foreseeable future and Kabongo Tshimanga relegated to the substitutes’ bench.
The main talking point early on surrounded ex-Pilgrim Nathan Stainfield, who caught United defender Ben Middleton with a flailing arm. Stainfield was booked and Middleton was left with a bloodied nose.
United had a let-off when Ashley Worsfold bobbled an effort into the arms of George Willis after seizing upon on a poor backpass from Andi Thanoj.
Willis denied Worsfold again moments later and Jarman’s follow-up flew wide with Stephen Brogan retreating.
United took the lead in the final minute of the first half when Bennett reversed a neat pass into the path of ROLLINS to angle home the game’s opening goal.
The Pilgrims added their points-cliching second goal in the 73rd minute when Reece Thompson and substitute Gregg Smith were denied in quick succession and ABBOTT rolled home the loose ball into an unguarded net.
Thanoj and Thompson were off target with late efforts for the visitors, but United were long since home and dry.
Boston (4-4-2): George Willis; Paul Walker, Ben Middleton, Jordan Keane, Stephen Brogan; Jay Rollins, Brad Abbott, Andi Thanoj, Ashley Hemmings; Reece Thompson, Lee Bennett (Gregg Smith 66). Subs (not used): Kabongo Tshimanga, Brad Beatson, Adam Curry, Jamie McGuire.
Gainsborough (3-5-2): Henrich Ravas; Ioan Evans (Josh Lacey 51), Michael Jacklin, Nathan Stainfield; Craig King, Jordan Richards, Shane Clarke, Liam King (Curtis Bateson 51), Tom Davie; Nathan Jarman (Alex Simmons 60), Ashley Worsfold. Subs (not used): Mason Warren, Ellis Storey.
Referee: Amy Fearn.
Attendance: 1,491 (69 visitors).