Manager Slaven Bilic said he was "angry and frustrated" after West Ham failed to reach the Europa League group stage.
The injury-hit Hammers lost to Romanian champions Astra Giurgiu in the qualifying round for the second season in succession.
The visitors won 1-0 in Thursday's second leg at the London Stadium to go through 2-1 on aggregate.
"I'm not going to lose faith in the players after one game," said Bilic. "We're missing top-quality players."
Former West Brom and Barnsley midfielder Filipe Teixeira's precise first-half finish secured victory for Astra as West Ham failed to convert any of their 17 attempts at goal.