City's best opening came when ex-Tiger Matt Hocking slipped, but Michael Branch's cut back was behind Gary Alexander and he was unable to get a shot in.
Most of the first half action involved the referee Pearson, booking four Pilgrim players and one Tiger.
Three minutes into the second half City took the lead in fortuitous circumstances. Damien Delaney mis-hit his left wing cross, and Paul Bastock was unable to get back to keep it out.
Things went from bad to worse for Boston minutes later when they went down to ten men. Michael Branch and Paddy McCarthy tangled in the box after Branch came in with a late tackle. Both were shown yellow cards, but for McCarthy it was his second so he was given his marching orders.
Stuart Green had an effort on goal well saved by Bastock to keep Boston in the match, before they went up the other end and had two chances of their own.
Mark Angel hit the post after slack defending on 56 minutes, before calling Paul Musselwhite into action with a shot two minutes later.
Substitute Stuart Elliott was twice denied with shots straight at Bastock, while Carl Regan was wasteful when through on goal midway through the half, firing waywardly off target.
Ian Ashbee shot just over in the 74th minute with a left foot drive from 20 yards out, while substitute Phil Jevons had a goal ruled out after finishing from Dean Keates' left-wing cross.
City kept hold of their 1-0 lead until the final whistle with Boston rarely threatening, but will not be pleased at the manner of the win.
City line-up (4-4-2): Musselwhite; Regan, Whittle, Anderson, Delaney; Green, Melton, Ashbee, Keates; Alexander, Branch.