Mark Noble fired the Hammers ahead from the penalty spot on the stroke of half-time following a handball but, shortly after Michail Antonio missed an opportunity to make it 2-0, Denis Alibec made them pay seven minutes from time.
West Ham also conceded late on in their opening Premier League match on Monday night, when Diego Costa netted a minute from time to secure a 2-1 win for Chelsea at Stamford Bridge.
But despite not winning the game in Romania, Bilic was content with a draw heading into next week’s second leg at the London Stadium.
“To be fair, they put us under pressure and had chances,” the Croatian told BT Sport.
“We had some bad clearances, not clean enough. Then they scored a good goal. It is a cheap one from our side.
“We were during the game good at set-pieces, especially in the first half from defensive ones. It is a shame, but still it is a good result.
“One-one, are we happy? Not totally. Is it a good result for us? We hoped that we were going to win but this kind of team mixed with some youngsters was a good experience. It is a good result for us.”
Bilic was disappointed with his side’s defending from the corner that brought the hosts’ equaliser, just after Antonio failed to put the game to bed.
“It is a shame because a couple of minutes before the goal when we had a clear-cut counter-attack,” Bilic added. “But unfortunately we missed the target.
“And instead of 2-0 and killing the game off or putting it to bed, they get a corner a minute later and score from it.”