On a disappointing night for the Hammers, Enner Valencia put them in front in the 23rd minute, only to then be stretchered off later in the first half.
Mauro Zarate made it 2-0 shortly after the break but Fernando Boldrin pulled a goal back for Astra after James Collins was dismissed.
The Romanians then levelled on the night through an Angelo Ogbonna own goal and Hammers boss Slaven Bilic was sent off late on.
Despite their negative approach to the tie, it was Astra who almost took the lead as former West Brom midfielder Filipe Teixeira saw his drilled effort whistle inches wide of Adrian’s goal.
But it was the Hammers who opened the scoring just moments later as Payet crossed for Valencia to nod home at the back post at the mid-way point of the first half.
The 25-year-old’s first game of the season came to a disappointing end as he was forced off injured, with Modibo Maiga on in his place only minutes after Bilic had been forced to shuffle his pack by introducing Reece Burke in place of the limping Joey O’Brien.
Payet was still the main attraction and he delivered a cross that Ogbonna headed towards goal, only to see Silviu Lung make a tremendous finger-tip save.
The Astra goalkeeper was at it again on the stroke of half-time, leaping even further to his left to push away Zarate’s curled attempt, but the Argentinian defeated him five minutes after the restart, finishing coolly after a jinking run.
Bilic then saw Collins dismissed, a regular sight for the manager in his early career in the Upton Park dugout, with Giurgiu twice coming close as the West Ham defence readjusted.
The visitors were still pushing and with 20 minutes remaining they gave themselves hope of springing an upset as Boldrin thrashed in a long-range effort off the underside of the crossbar.
The equaliser came in unfortunate fashion as Ogbonna hurled himself at a hopeful cross, only to divert the ball past a despairing dive from Adrian.
More controversy came as Maiga’s claims for a last-gasp penalty fell on deaf ears, with Payet booked for his remonstrations and Bilic sent from his technical area for his own complaints – the Croatian also knowing his side have it all to do next week.