Tony Thorpe started the game in place of the injured Rory Fallon, long with David Stroud making his home debt after playing in last weeks 1-1 draw with Huddersfield.
After Swindon had a penalty appeal turned down on 24 minutes Boston took the lead a minute later with a shot from 35 yards out by Simon Rusk beating Tom Heaton in the Swindon goal.
Swindon equalised 7 minutes later after Miglioranzi was fouled outside the box Andy Gurney took the free kick and scored from 30 yards out.
Just before half time Boston took the lead again with another long distance shot, this time from 25 yards out from Stewart Talbot which struck the post, rebounded out hitting Tom Heaton, and rolling very slowly over the line.
In the second half Miglioranzi came close to getting an equaliser, but his shot was saved. Swindon then had a couple of corners within a minute one which resulted in Sean O'Hanlon beating the 'keeper, but having his shot headed off the line by a Boston defender.
Boston had very few chances to score any more goals in the second half as Swindon bought on Charlie Comyn-Platt to replace Miglioranzi for the final 7 minutes of normal time.
Then on 87 minutes a ball from Gurney found Comyn-Platt who scored his first goal for the club from close range after a mix up between the Boston 'keeper and one of his defenders.
Swindon then pushed forward for the winner, and came close a number of times in the final few minutes, but go into the draw for the second round thanks to a last minute save from Tom Heaton saving a one of one with Boston's Danny Thomas.
Overall a draw was a fair result with it's being a fairly average game overall. Boston only had three chances to score, and they took two of them, and Swindon's chances in front of goal were also limited. Swindon must now travel up to Lincolnshire for a replay within the next couple of weeks.