Astra, who drew the first leg 2-2 in London having trailed by two goals, completed a 4-3 aggregate victory thanks to a double strike in four first-half minutes from their captain Constantin Budescu.
Slaven Bilic's West Ham, who had qualified for the competition through the Fair Play League, had gone 1-0 up when debutant Manuel Lanzini scored after only three minutes.
But they fielded an inexperienced team, having made 11 changes to the side from the first leg, and were picked off when Budescu struck in the 32nd and 36th minutes to end West Ham's first European campaign since the 2006-07 season.
Premier League Southampton cruised through to the playoff stage 5-0 on aggregate after a 2-0 win at Dutch side Vitesse Arnhem.
(Reporting by Toby Davis, editing by Pritha Sarkar)
