A solid rear-guard display together with two early goals from Watford loanee Albert Jarrett finally held sway over a second-half reply by Rover's substitute Lewis Haldane.
It took Boston just two minutes to break the deadlock.
Jarrett stepped up to convert a penalty, awarded after Sam Oji had handled in the area when attempting to clear a deflected free-kick by Jamie Clarke.
And Jarrett doubled Boston's opening delight in the ninth minute when he curled a 20-yard left-foot drive beyond Steve Phillips, after being set free by Nathan Joynes.
Between Boston's goals, Rovers had come close to an equaliser but Craig Disley saw his diving header from a Sam Igoe cross bounce away off the upright.
After a spell of ebb and flow play, it was Boston who almost scored again. Drewe Broughton saw his 27th minute header bounce off the bar from a Jarrett free-kick.
The visitors found a degree of attacking impetus in the 38th minute and Chris Lines got into a shooting position but saw his deflected 20-yarder clutched by keeper Andy Marriott.
The home keeper then made a tremendous save seconds before the interval when dashing out to block a Joe Jacobson drive after the full-back had been put clear by Disley.
Five minutes after the break Boston again went close. Broughton seeing his lofted effort clip the outside of a post, after Joynes had flicked on a long clearance by Marriott.
Rovers pulled a goal back in the 68th minute when Igoe's cross from the right was only partially punched away by Marriott. This allowed Chris Carruthers to deliver a low-centre for substitute Haldane to stab in from eight yards.
That goal signalled a spell of pressure by the visitors but further assaults were repelled by a home rear-guard which showed good resolve in the face of a rising Rovers corner tally as the minutes ticked away.