Jordan Nuttell handed the Pilgrims an early lead, but once Luke Weaver had nodded the Red Imps level on the stroke of half-time, they appeared to be only one winner.
Second half goals from Alex Simmons and Aaron Rice capped Lincoln’s comeback.
Lincoln had started well at The Jakemans Stadium, with Sam Vince called upon to deny Jordan Thomas after only two minutes.
United almost took the lead on the 10-minute mark. Nuttell forced the ball past a back-peddling Ollie Pannell, but it stopped dead on the goalline before a United player could force it home.
Boston did take the lead four minutes later. Charley Sanders whizzed over a right-wing cross and NUTTELL headed home from close-range.
Joe Briers and Robert Taylor had respective efforts deflect wide, while Vince excelled to deny Simmons from distance.
The Imps levelled on the stroke of half-time when WEAVER glanced the ball home following Taylor’s right-wing delivery.
Ex-Pilgrim Will Rawdon missed a good chance for the Imps on the stroke of half-time, while Simmons spurned an even better one-on-one chance deep into stoppage time.
Lincoln’s Thomas clipped the top of the crossbar at the beginning of the second half and the Red Imps were soon in front (on 58 minutes) when SIMMONS blasted home after Thomas had dragged an attempted shot into his path.
The visitors capped their victory five minutes later when RICE rammed the ball home despite two initial outstanding saves from Vince.
United’s second and final group game will take place against Chesterfield at the Proact Stadium on Friday November 23.
Boston (4-4-2): Sam Vince; Niall Delahoy, Charlie Melville, Josh Spencer, Todd Devlin (Jack Hall 22); Curtis Revell, Aaron Eyett, Keiron Elliott (Lewis McKue 77), Joe Briers; Jordan Nuttell, Charley Sanders (Leon Squire 64). Subs (not used): Juris Kormis, Tom Benton (gk).