United endured a frustrating first hour but edged ahead through Abbott on 64 minutes and doubled their lead through a fantastic solo effort from Thewlis two minutes later.
Avenue forced a frenetic finale when Brad Dockerty reduced the arrears with six minutes to play, but United held on.
Pilgrims’ boss Craig Elliott named an unchanged starting XI, with fit-again George Green replacing the ill Tom Clare on the bench.
Visiting defender Nelson Digbeu flattened Jay Rollins early on down the United right and from the resulting quick free-kick, Abbott crashed an effort marginally wide.
Former Pilgrim Ryan Cresswell, playing up front for Avenue, had two opportunities in quick succession for the visitors, but was unable to steer either effort on target.
With the match meandering after the break, Elliott introduced Thewlis and fellow substitute Jordan Adebayo-Smith and the response was almost immediate.
Adebayo-Smith and Alex Whittle combined on the United left and when Avenue were unable to clear their lines, ABBOTT swooped to bury a low effort to hand Boston the lead.
THEWLIS subsequently got in on the act by waltzing through the Avenue rearguard to beat Jack Bennett at this near post with a fine driven finish.
The Pilgrims were seemingly comfortable but were made to sweat in the closing stages when substitute Myron Gibbons nipped in front of Peter Crook to tee up fellow sub DOCKERTY to walk the ball home.
Riccardo Calder came within inches of a dramatic equaliser from 20 yards, but United remained resolute and saw the job through.
Boston (4-3-3): Peter Crook; Karl Byrne, Luke Shiels, Simon Ainge, Alex Whittle; Brad Abbott, Andi Thanoj (Jordan Adebayo-Smith 56), Tom Platt; Jay Rollins, Dominic Knowles (George Green 69), Nicky Walker (Jordan Thewlis 56). Subs (not used): Ashley Jackson, George Willis (gk).
Bradford (4-5-1): Jack Bennett; Isaiah Bazeley-Graham, Mitchell Lund, Nelson Digbeu, Riccardo Calder; Lewis Knight (Jaheim Headley 31), Isaac Marriott, Marcus Wood, Alex Hurst, Elliot Holmes (Brad Dockerty 62); Ryan Cresswell (Myron Gibbons 82).
Referee: Scott Simpson.
Attendance: 1,110 (47 visitors).