However, City gave away their lead 10 minutes into the second half with a soft goal. Neil Redfearn sent in a free kick from the left wing to the back post, and Paul Ellender met it with City's defence caught asleep.
City were on the offence straight from the restart and Stuart Green went inches wide with a shot.
The turning point came in the 67th minute when ex-Tiger Matt Hocking was given his marching orders. Allsopp was running on to a through ball and last man Hocking hauled him down, giving the referee no option but to send him off. Boston brought on another former City defender in the form of Mark Greaves, and survived the resulting free kick from Hocking's foul.
Peter Taylor sent Jamie Forrester on in place of Alton Thelwell as City looked to make their numerical advantage count by playing three strikers. Midfielders Dean Keates and Steve Melton also came on to freshen up a City side who were struggling to break down Boston's defensive stance.
But it wasn't until the final minute that City made the breakthrough. Damien Delaney crossed from the left and Stuart Green met it with a header into the bottom corner of Bastock's goal. The midfielder ran to the touchline and was booked for kicking a corner flag in two in celebration of the three points.
City line-up (4-4-2): Musselwhite; Thelwell (Forrester 76), Whittle, Hinds, Delaney; Price, Green, Ashbee, Elliott (Keates 81); Burgess (Melton 85), Allsopp.
Un-used subs: Kuipers, Holt.