The Spanish striker rescued his side by netting just his second goal of the season, and his first in seven games, to break down the visitors' resolute defence.
Bournemouth held their own during the opening three quarters of the match in West London, but clinical strikes from two of Chelsea's flair players made the difference
Arrizabalaga arrived at the club for a world record fee for a goalkeeper of 80 million euros (71.89 million pounds) from Spanish side Athletic Bilbao.
The Blues dominated the early stages of the encounter, with Ross Barkley forcing a save from Samir Handanovic three minutes in and Alvaro Morata seeing a headed goal ruled out for offside shortly after
However, there was no place for Chelsea striker Morata, with Isco, Marco Asensio, Lucas Vazquez, Rodrigo and Diego Costa preferred in attack.
Chelsea striker, Alvaro Morata, was left out after a disappointing season in England as was his team mate Marcos Alonso, with Arsenal’s Nacho Monreal picked as back-up for left back Jordi Alba.
Giroud muscled and dribbled past three defenders and keeper Alex McCarthy seconds into the second half to poke the ball home from a pass by the ever-busy Eden Hazard.
West Ham, who had been much improved in beating Southampton 3-0 a week ago, offered little until Hernandez’s intervention soon after coming on.