Paul Bastock didn't have too many challenging saves to make, but was beaten in the 19th minute for the opening goal. Stuart Elliott drove a low free kick into the bottom corner for his 12th of the season.
Bastock saved from Danny Allsopp to keep the score down to 1-0 at half time, and City were perhaps disappointed they hadn't made more of their chances to increase that lead.
After Mark Greaves had one disallowed for Boston, City managed a second in the 63rd minute. Myhill's long ball was headed on by Ben Burgess to Allsopp who finished left-footed past Bastock.
But Boston pulled it back to 2-1 just minutes later. A long throw-in wasn't dealt with and centre back Paul Ellender fired in.
The home side piled on the pressure for an equaliser, but frequently left themselves exposed on the counter attack. Ben Burgess, Stuart Green and substitute Jon Walters all had chances to make the win safe but failed to capitalise. However, it wasn't to matter as City held on for the three points to go just one behind Doncaster, and also have a game in hand over the league leaders.
City line-up (4-4-2): Myhill; Thelwell, Joseph, Hinds, Delaney; Marshall (France 74), Green, Lewis, Elliott; Allsopp (Walters 75), Burgess.
Un-used subs: Musselwhite, Holt, Forrester.