The Poachers put aside poor league form to book a quarter final Knockout Cup derby at Holbeach.
Jordan Nuttell opened the scoring for the visitors after 25 minutes only for Brian Farrell to level three minutes later.
Nuttell regained the lead for the Poachers in the dying seconds of the first half.
Leon Simpson scored a second Sileby equaliser after 53 minutes, but within two minutes Richard Ford had put Boston back in front.
A bizarre 70th minute own goal took the tie into extra time as visiting goalkeeper James Doughty could only fumble a Lewis Burgess long throw into the net.
Craig Parker put Boston ahead again four minutes into extra time and Alex Beck sealed their last eight place in the last minute of the extra thirty.
Town: Doughty, J Ashton, Field, Ford, Stevens, S Ashton, Callaby (Fleming), Dykes, Beck, Nuttell, Parker