Krisztian Gagyi's 22nd minute strike handed the Pilgrims a half-time advantage.
Altrincham equalised through a Taylor Redman own goal with 20 minutes to play.
Boston (4-2-3-1): Cameron Fox; Taylor Jaine, Taylor Redman, Owen Sheriff, Lucas Stubbs; Jake Henderson, Jack Waddington (Rohann Boughey 37); Krisztian Gagyi, Han Stevens, JJ Ward (Jake Higginbottom 70); Flynn Slater (Josh Doran 78).
Referee: Ian Ruddock.