Elliott struck in the fourth minute of added time to save the Lincolnshire Senior Cup tie heading to a penalty shootout.
Matt Tootle on 32 minutes and Jordan Burrow two minutes before half-time had twice handed the lead to the visitors, but the Whites responded through Andy Janssen’s fine 41st minute strike and Callum Foster’s scrambled effort from a 75th minute corner.
Boston (4-4-2): Peter Crook (Josh Barnes 46); Matt Tootle, Scott Garner, Andy Butler (Luke Shiels 56), Scott Duxbury; Fraser Preston (George Hornshaw 66), Paul Green (Shane Byrne 56), Connor Dimaio (Tom Platt 66), Joe Leesley (Terry Hawkridge 56); Jordan Burrow (Danny Elliott 56), Jordan Preston.
Referee: David Jones.
Attendance: 222.