Callum Gittings scored the only goal of the game with 10 minutes remaining after Henrich Ravas had failed to hold a ball into the box.
Moors were indebted to a tremendous save from Aston Villa loanee Benjamin Siegrist in stoppage time to deny substitute Kyle Dixon and end any United hopes of a replay.
United arguably deserved much more for a purposeful display on the home ground of their unbeaten league rivals, but one slip proved decisive.
Both sides rattled the post during the first half, with Adam Walker and Dayle Southwell both hitting the woodwork from distance, while Southwell had earlier fired wide following Grant Roberts' low corner.
Carl Piergianni's header was cleared off the line by Ryan Beswick following another Roberts corner.
More great play from Roberts put the recalled Ryan Robbins away down the United right but his finish was lacking after weaving inside Gavin Cowan's challenge.
United continued to create after the break, with Zak Mills' rising drive clearing the bar following Roberts' partially-cleared corner, while excellent play from substitute Cameron Johnson played Southwell through and his drive was beaten aside by Siegrist.
Jordan Blinco's deflected drive then forced Siegrist into a scrambling save before Roberts motored brilliantly to play in Southwell, but United's top scorer was only able to bend a shot wide.
The game's defining moment arrived in the 80th minute when Ravas failed to hold Steven Leslie's cross and GITTINGS turned the ball home on the line.
Beswick was inches away from killing the tie six minutes later when he rattled the bar from distance but it was left to Siegrist to stick out a big right leg to divert Dixon's goalbound drive behind for a corner to prevent a replay.
Solihull (3-4-1-2): Benjamin Siegrist; Theo Streete, Liam Daly, Gavin Cowan (Callum Gittings 66); Junior English, Adam Walker (Steven Leslie 77), Ryan Beswick, Connor Franklin; Darryl Knights, Akwasi Asante, Stefan Moore. Subs (not used): Gareth Dean, Jack Byrne, Darren Acton (gk).
Boston (4-4-2): Henrich Ravas; Liam Marrs, Zak Mills, Carl Piergianni, Dylan McEvoy; Kaine Felix (Jay Rollins 61), Jordan Richards, Grant Roberts, Jordan Blinco (Kyle Dixon 82); Dayle Southwell, Ryan Robbins (Cameron Johnson 53). Sub (not used): Jason Alexander (gk).
Referee: Anthony Tankard.
Attendance: 319.