Goals from Andy Gascoigne and Ryan Fallowfield eased the pressure on boss Simon Weaver and took Town up to 16th, seven points above the relegation zone.
The match survived two pitch inspections in the morning as snow covered the pitch.
Weaver handed a surprise debut to Goole midfielder Cecil Nyoni, who he signed on Saturday morning.
However, it was Nyoni’s former Goole teammate Gascoigne that gave Town the perfect start when he slammed home Fallowfield’s cross on 11 minutes.
Teenage Hull City loanee Fallowfield then doubled the home side’s advantage as he rifled home Paul Clayton’s cross from close range.
Town looked to be cruising towards victory until Boston’s Mark Jones late header gave the visitors a glimmer of hop.
Goalkeeper Peter Crook then heroically denied United striker Marc Newsham deep into injury-time to ensure the points stayed at the CNG Stadium.