The 20-year-old, who thrived on loan at Derby County as they narrowly missed out on promotion to the Premier League via the play-offs last season, is looking to force his way into Jose Mourinho’s plans.
And he did his cause no harm with a sweet volley to open the scoring from Nathan Ake’s left-wing cross early on.
Teenage defender Kurt Zouma came on for the injured Todd Kane before Bamford had a good chance to finish off a neat Chelsea move just before the break.
But it took the first of two well-taken goals from the promising Izzy Brown to get the Blues moving in the second half – his first coming from a good cross from first-year scholar Jay Dasilva.
Club captain John Terry, who signed a new one-year deal at the end of last season, blasted in from a corner before seasoned defensive partner Branislav Ivanovic headed in the fourth, and Brown added the fifth.
Chelsea will play AFC Wimbledon at Kingsmeadow in their next pre-season friendly on Saturday before they depart on a tour of Europe.
They will take Bamford with them in order to make a full assessment of whether he can push Diego Costa and fellow Spaniard Fernando Torres for a place up-front this season.