United trailed Stourport Swifts 2-0 at The Jakemans Stadium, but two goals apiece from Stuart Whitaker and James Reed got the Pilgrims in front, before the visitors ensured a nervy finale with a late third.
The Pilgrims will now travel to Burton Albion during the week commencing October 31.
United's fourth qualifying tie of the campaign promised to be their hardest task to date and Stourport were bright from the off.
With newcomer Josh Macann slotting in behind a defence which had not conceded in the competition prior to kick-off, United took the game to their visitors.
Striker Charley Sanders, who had sustained an injury during the warm-up, soldiered on and had one shot blocked, before forcing keeper Jordan Andrews into an excellent low save from 18 yards.
Macann had to make a save to keep out a decent effort from Billy Halley, while Jack Cullity curled an effort narrowly wide for the away side.
Sanders had another header blocked but eventually succumbed to the knee problem picked up before kick-off, with the speedy Joe Briers deputising from the bench.
The Pilgrims should have taken the lead when Whitaker delivered a great ball from the left and the usually-reliable James Tricks slammed a close-range effort over the bar.
United were punished within four minutes. CULLITY was afforded too much space down the Stourport left and he picked his spot, low to Macann's right, from 20 yards.
The Pilgrims' woes deepened a further three minutes on. Cullity seemed certain to add his second, but fluffed his kick in front of goal, only for Connor SMITH to arrive to slam the loose ball past a stranded Macann.
Swifts' keeper Andrews made a great save to keep out a Tricks drive, following Briers' cross, as the Pilgrims sought an instant riposte. Briers then had a firm 16-yard drive beaten away by Andrews.
The game's defining moments arrived in first half stoppage time. Smith was a whisker away from adding Swifts' third, only for WHITAKER to crash home a neat volleyed finish on the stroke of half-time to ensure the deficit was cut to one.
The second half was seven minutes old when United levelled. Tricks was the instigator down the left and REED applied a fabulous finish from 18 yards.
United moved in for the kill and Tricks' effort was ruled out for offside, after centre-half Lewis Sturman had rattled the crossbar.
The Pilgrims edged in front on the hour mark. Whitaker and Briers combined, before the latter played a neat one-two with REED, who in turn crashed home a 20-yard beauty off the underside of Andrews' crossbar.
Substitute Jack Roe wasted two presentable chances for the visitors, while Macann held a firm drive from Luke Griffiths.
Briers fired an effort too high for Boston, but the Pilgrims were soon two goals to the good. Briers and Tricks combined brilliantly down the left and the latter's cross was headed home at the back post by WHITAKER.
The game was seemingly won, with Swifts looking a dejected outfit, but they were given hope nine minutes from time. Slack defending from Boston allowed ROE space and he finally did what he had been threatening to do by cutting the deficit with an excellent low finish.
Stourport piled bodies forward in search of an equalising goal which would have taken the tie to extra-time and with Boston unable to retain possession, a nervy ending followed.
The visiting contingent thought salvation had arrived in the 89th minute when keeper Macann flapped at a deep cross, Sturman dived in and substitute Jonathan Jenkins went tumbling, although referee Duncan Robertson awarded a goal-kick as the visitors' claims for a penalty were waved away.
Boston (4-4-2): Josh Macann; Niall Delahoy (Sam Towell 80), Lewis Sturman, Jordan Nuttell, Charlie Rossington; James Reed, Elliott Price (Jake Platten 69), Jake Beauchamp, Stuart Whitaker; Charley Sanders (Joe Briers 19), James Tricks. Subs (not used): Scott Langan, Alex Marr.
Stourport (4-4-2): Jordan Andrews; Jack Fields (Jack Roe 56), Toby Jeffries, Rhys Tercana, Dominic Wheeler; Connor Smith, Sam Keltch (Nick Pritchard 83), Ben Hayes, Jack Cullity; Billy Halley (Jonathan Jenkins 68), Luke Griffiths. Subs (not used): Andrew Hudson, Lawrence Clements.
Referee: Duncan Robertson