The Hammers are sorely missing the attacking talents of their injured stars Dimitri Payet, Diafra Sakho, Manuel Lanzini and Victor Moses in front of goal.
Yet Bilic, who has ordered the club to move training grounds in a bid to halt their injury jinx, would have seen his side pick up a first win in six matches had Mauro Zarate's 80th-minute shot not cannoned back off a post.
A draw was a fair result, though, as Stoke were also denied by the woodwork when Marko Arnautovic hit the crossbar midway through the second half.
"I'm pleased on one hand, considering we are missing key offensive players, with the way we played," said Upton Park boss Bilic.
"If you ask me before the game if I'm happy to take a point, then no
But considering the way the boys played, I can't be unhappy
I couldn't ask for more."
Stoke should have taken an early lead but Marco van Ginkel scuffed a sitter wide from in front of goal.
As well as the woodwork, both goalkeepers were kept busy with Jack Butland making fine saves from Cheikhou Kouyate's long-range effort and Andy Carroll's header, while Adrian twice denied Stoke sub Meme Biram Diouf with his legs.
And in stoppage time a frantic goalmouth scramble saw Kouyate's header cleared from underneath the bar by former West Ham defender Glen Johnson
"It was a very entertaining game, full of energy, high tempo with both teams defending well," added Bilic
"It should have been 1-1 or 2-2
With a bit of luck we could have one, but so could they.
"They also wanted to win
The game deserved goals."
The Hammers are now without a win in six matches while Stoke, fresh from last weekend's victory over Manchester City, have lost just one in six.
"It was a close game," said Potters manager Mark Hughes
"For a 0-0 it had plenty of incident
Someone told me there were 30-odd attempts on goal from each side.
"West Ham were a bit more direct because of all their injuries and we stood up to that.
"West Ham did a good job, they knew what we did last week and they weren't gong to allow that to happen at home.
"It was a decent game to be involved in and we'll look back at it at the end of the season and say it was a decent point."
Source : PA
Source: PA