Hosts AFC Telford United had led courtesy of Shane Sutton's 29th minute strike and despite the same player inadvertently putting through his own net five minutes before the break, the Bucks prevailed courtesy of Dwyer's long-range stunner.
The result extended the Pilgrims' winless run away from home in Vanarama National League North.
It was a quiet opening in Shropshire, with both sides struggling to create clear-cut chances and in truth there was nothing noteworthy until Telford edged ahead.
Ryan Wilson delivered a deep corner from the Telford right and SUTTON arrived at the back post to guide the ball past George Willis after shrugging off Brad McGowan.
John Marsden had an effort from distance which swerved off target at the last moment before the visitors equalised when Jack Thomas’ cross from the right was headed into his own net by SUTTON.
The Pilgrims thought they were going to reach the interval all-square but DWYER had other ideas after seizing upon a loose ball in midfield and depositing a fantastic effort beyond Willis’ grasp from fully 30 yards.
The visitors’ anticipated onslaught after the break struggled to gather momentum and it was Telford who went close again with Sutton rattling the crossbar and Willis saving well to deny Marsden.
Substitute Gregg Smith had a header drop wide following a long Willis free-kick but United found opportunities hard to come by as the points slipped through their grasp.
AFC Telford (4-2-3-1): Jasbir Singh; Ross White, Shane Sutton, Connor Johnson, Ryan Wilson; Henry Cowans, Jonathan Royle; Anthony Dwyer (Jordan Lussey 86), John Marsden (Louis McGrory 90), Elliott Newby; Marcus Dinanga (Jordan Murphy 72). Subs (not used): Jordan Gough, Sheridan Martinez (gk).
Boston (4-3-3): George Willis; Liam Wakefield, Brad McGowan, Shane Killock, Jan Yeomans; Jordan Keane (Liam Adams 46), Jamie McGuire, Jack Broadhead; Jack Thomas, Kabongo Tshimanga (Gregg Smith 67), Jay Rollins (Ashley Hemmings 75). Subs (not used): Adam Chapman, Karl Hawley.
Referee: Declan Bourne.
Attendance: 1,097.