Luke Shiels scored the only goal of the game in the 69th minute as the Pilgrims avoided a potential banana skin.
Dominic Knowles was denied from the penalty spot in stoppage time, while the Royals finished with 10 men when Desean Martin was sent off for punching Tom Platt.
The home side started well on their 3G pitch, with Peter Crook required to save at the feet of Jordan Francis while Shiels produced a fantastic block to deny Ryan Shaw after Isai Marselia had charged down Crook’s clearance.
United threatened meaningfully for the first time when Nicky Walker threaded the ball through for Jordan Thewlis to arrow an effort narrowly wide of the far post.
Alex Whittle then teed up Brad Abbott, whose effort deflected over the bar while Simon Ainge had an effort scrambled off the line following Walker’s corner.
Tom Platt cut infield to drill an effort over the bar on his left foot as United built up a head of steam before half-time.
Home keeper Lewis Gwilliams tipped a Knowles effort wide at the beginning of the second half before United’s breakthrough arrived with 21 minutes to play.
Walker delivered a pinpoint corner delivery and SHIELS bulleted home a close-range header.
United had further chances to kill the contest with Knowles firing into the side-netting following Brad Abbott’s assist before the Pilgrims won a stoppage time spot-kick.
Shaun Tuton was upended by James Beresford and Knowles’ low penalty was gathered by Gwilliams.
Sutton Coldfield went straight up the other end and Edwin Ahenkorah forced Crook into a save before all hell broke loose in front of the dugouts.
Martin swung a punch at Platt and was eventually dismissed when tempers had cooled and United saw out the remaining seconds to secure much-needed cup progress.
Sutton Coldfield (4-4-2): Lewis Gwilliams; Vinnie Dwyer, James Mutton, James Beresford, Aris Christophorou (Joel Powney 83); Ryan Shaw (Edwin Ahenkorah 67), Desean Martin, Joe Halsall, Jordan Francis (Chinendu Okoro 54); Reece Leek, Isai Marselia. Subs (not used): Reece Gibson, James Harrison (gk).
Boston (4-4-2): Peter Crook; Ben Middleton, Luke Shiels, Simon Ainge (Lewis Gibbens 60), Alex Whittle; Jordan Adebayo-Smith (Shaun Tuton 87), Brad Abbott, Tom Platt, Nicky Walker; Dominic Knowles, Jordan Thewlis (Jonathan Wafula 78). Subs (not used): Andi Thanoj, Ashley Jackson, Karl Byrne, Jay Rollins.
Referee: Jacob Miles.