The Pilgrims were already a goal down at half-time courtesy of James Armson's 44th minute strike when Davies handled on the line.
Shane Byrne doubled the Saints' lead from the subsequent spot-kick as Brackley coasted to a comfortable victory.
The Pilgrims, missing four players through injury, made the early running at St James Park and were unlucky not to find themselves ahead in the opening stages.
Home keeper Danny Lewis made fine saves to keep out Brad Abbott and George Smith, either side of almost fumbling Smith’s cross from the left into the net, with Connor Hall sparing his blushes on the line.
Ben Middleton’s header was hacked off the line by Glenn Walker following Nicky Wroe’s deep corner to the back post.
Brackley took the lead a minute before half-time when Lee Ndlovu strode confidently away from Middleton and clipped the ball across for ARMSON to finish emphatically from eight yards.
George Willis made a solid stop to deny Adam Walker at the start of the second period but the Saints were soon two goals to the good.
Hall’s header was handled on the line by Davies and with the United full-back duly red-carded, BYRNE sent Willis the wrong way from the spot to double Brackley’s advantage — a lead they never looked like relinquishing.
Brackley (4-2-3-1): Danny Lewis; Shepherd Murombedzi, Gareth Dean, Connor Hall, Glenn Walker; Shane Byrne, Adam Walker; Matt Lowe (Ellis Myles 75), James Armson (Daniel Nti 82), Carl Baker (Luke Fairlamb 63); Lee Ndlovu. Subs (not used): Oran Jackson, Omari Sterling-James.
Boston (4-5-1): George Willis; Ben Davies, Ryan Qualter, Ben Middleton, Ashley Jackson; Max Wright (Andre Johnson 83), Brad Abbott (Cameron Hawkes 85), Nicky Wroe (Tom Clare 76), Andi Thanoj, George Smith; Jay Rollins. Subs (not used): Jordan Slew, Dylan Parkin (gk).
Referee: Matt Corlett.
Attendance: 755.