The Pilgrims never recovered from Elliot Powrie's second minute strike - with Divine Iyamu doubling the visitors' lead in the 27th minute.
Patrik Bad was later dismissed for two yellow cards, with Harvey Gotheridge wrapping up Eastwood's fine win in stoppage time.
Boston: Dan Martin, Reece Coy, Patrik Bad, Will Langford, Charlie Brooks, Finley Ging, Rhys Smaller, Rueben Fogg, Charles Doble (Josh Davenport 46), Zac Hardy (Sean Bray-Smith 83), Toby Cotton (Taylor Redman 46). Subs (not used): Dexter Tomlinson, Lucas Stubbs, Finley Hipkin, Austin Barker (gk).
Referee: Aimee Keir.
Attendance: 222.