Goals from Enner Valencia and Mauro Zarate had given the Hammers a seemingly comfortable lead, though James Collins was sent off for a second yellow card in the 59th minute, prompting a remarkable comeback.
Rick Boldrin pulled a goal back with a stunning 71st minute strike from range, before Angelo Ogbonna netted an embarrassing own goal with eight minutes to play to level the scores.
It leaves new West Ham boss Bilic - who himself was sent to the stands late on - with plenty to ponder ahead of the Premier League season with the injury-enforced withdrawals of Joey O'Brien and Valencia adding to a frustrating evening at Upton Park.
As part of a dominant first-half display, the hosts made much of the early running with lively debutant Dimitri Payet showing why West Ham paid an eight-figure fee for him.
Zarate - seemingly brought in from the cold by Bilic after a loan spell at QPR last term - saw two shots blocked in quick succession as West Ham applied further pressure, although they were caught napping when Astra midfielder Filipe Teixeira found the side-netting following a quick break.
However, Payet helped give West Ham the lead when he crossed for Valencia to power home at the far post.
Despite Valencia and O'Brien being forced off, West Ham continued to threaten after the break and Zaratedoubled their advantage by jinking past two Astra defenders before beating Silviu Lung to give his side some breathing space.
The Argentine was thwarted in his search for a second as things seemed to be heading to a natural conclusion for the hosts, before Collins picked up the second of his bookings and joined Diafra Sakho and James Tomkins in seeing red in Europe.
Skipper Budescu found the top of the net with the resulting free-kick while Boldrin's corner was flicked on and blocked on the line by Mark Noble as the visitors sought to press home their numerical advantage.
And when Boldrin was afforded time and space to shoot, the Brazilian found the net from distance via the underside of the crossbar to give Astra an away goal to return home with.
However, it was to get better for Marius Sumudica's men when Ogbonna tried to clear a low cross, only to shin the ball past Adrian, levelling matters ahead of the second leg next week.
