The Young Lions batted first in Cox's Bazar, reaching 209/7 from their 50 overs. Goldsworthy was his team's top-scorer, while there were also useful contributions from Middlesex's Luke Hollman (35*) and Yorkshire's George Hill (30).
In reply an early wicket for the visitors - courtesy of a run-out - kept them in the hunt but the hosts batted sensibly and without too much alarm to chase down England's total for the loss of five wickets with five overs to spare.
Mohammad Parvez Hossain top-scored for Bangladesh (80), while for the Young Lions there were wickets for Hollman, Tom Lammonby, Adam Finch and Hamidullah Qadri.
The second match of the 50-over series takes place at the same venue on Thursday