Slaven Bilic fielded a youthful skeleton squad which included competitive debuts for no fewer than three players.
One of those was the club's latest signing, Argentine Manuel Lanzini.
In the build-up to the game we tipped new signing Lanzini to be West Ham's Europa League hero after shining when he came on a second half sub against Werder Bremen last time out.
And it took him just three minutes to start proving that prediction right that with a brilliantly taken goal as he tricked his way inside the corner of the 18 yard box before bending a shot in at the near post to give the Hammers a 1-0 lead.
It was a goal their bright start deserved and came just moments after captain Kevin Nolan had almost put Slaven Bilic's side 1-0 up.
Astra Giurgiu grew into the game, though, and could have been level on 15 minutes when Randolph continued his inauspicious start to life at West Ham by completely misjudging a free-kick and allowing it to go over his head only for it to hit the upright.
Randolph got lucky on two counts in that it hit the post but also just missed ricocheting off him and going into his own net.
There were hearts in mouths moment on 19 minutes when Diego Poyet was lucky to avoid a booking after a late challenge and plenty of theatrics from the opposition.
The Hammers did not continue their run of red cards, though, as the referee chose to warn Poyet of his future conduct.
But West Ham's good work was undone when Astra's star man Budescu turned the match on its head in five minutes.
On 31 minutes the number 10 was allowed far too much space as Doneil Henry backed off too far and Budescu let fly from 25 yards through the Canadian defender's legs into the bottom corner.
Randolph did not get down quick enough, either and should have done better.
Five minutes later it was 2-1 on the night and 4-3 on aggregate when Budescu arrived in the box unmarked to slot home unchallenged as Poyet failed to stop his man getting into the box and squaring the ball back for another gift of a goal.
The second goal seemed to knock the stuffing out of Bilic's inexperienced side which included three young debutants.
But in truth it does not change the tie too much with the Hammers only needing one goal to force extra-time as both sides will have two away goals.
Astra Giurgiu 2 West Ham 1 (Astra lead 4-3 on aggregate)