The Stags had dominated the first half before falling behind against the run of play in the 49th minute. But three goals in 17 minutes seemed to set them up for a comfortable victory.
However, Boston almost snatched an unlikely point in a thrilling finish. As the game entered injury time substitute Lee Thompson slid in a 16-yard finish from Chris Holland's clever pass.
Then just 30 seconds later Thompson again raced clear only to roll his shot narrowly wide with keeper Jason White stranded.
Earlier Mansfield boss Carlton Palmer had given another teenager his full first-team debut and 18-year old striker Danny Herron almost made an immediate impact.
In the tenth minute his header released Simon Brown whose pass allowed Alex Neil to cross for Richie Barker to head back to Giles Coke, but the young midfielder poked his finish just wide.
Three minutes later Brown was denied by the woodwork when his glancing header from a Neil corner clipped the bar.
The Stags carved out another clear chance in the 14th minute. Neil's pass beat the offside trap but Barker fired tamely at keeper Nathan Abbey's legs.
Injured referee Mike Pike swapped places with fourth official Steve Bratt at half time and his first decision proved contentious when he allowed Courtney Pitt to race clear despite a clear handball and cross for Bradley Maylett to give Boston a surprise lead four minutes into the second half.
Sixty seconds later another fine Pitt cross found half-time substitute Daryl Clare, but his tame shot went straight to keeper White.
And that miss proved vital. In the 53rd minute Brown's close-range shot was only parried by Abbey and Barker pounced to level with his tenth goal of the season.
Twelve minutes later substitute Callum Lloyd fired in from the edge of the area after a clever turn.
Within 60 seconds the teenage midfielder should have scored again, but this time he skied a Barker cross over the bar from six yards.
However, that miss didn't prove vital as Adam Rundle's side-footed 25-yard volley clinched victory in the 70th minute despite the injury time drama.