Stoke were heading for their first away win since October after substitute Peter Crouch capitalised on a mistake from keeper Joe Hart after 79 minutes to tap home from close-range.
Victory would have moved second-bottom Stoke above Southampton and three points from the safety zone but they failed to hold on and Andy Carroll came off the bench to show his quality with a superb volley past Jack Butland.
Despite dropping two crucial points, Lambert said he still took a number of positives from the result and felt his team would stand a fighting chance of beating the drop if they beat Burnley at home on Sunday.
“I thought we deserved something. I thought we played a really good game and if we keep playing like that, we will win games,” Lambert told a post-match news conference.
“But we have been playing really well of late. I keep saying it – if there is a little bit of Lady Luck, it has certainly not come our way. But performance-wise it was very good.
“I think if the team had been playing this way from August, I wouldn’t be here and they wouldn’t be in the position they are in. West Ham turned over Stoke a few months ago. They are still looking for points as well. With the atmosphere in stadium, it is a difficult place to come but I thought we were excellent.”
The draw leaves Stoke on 28 points from 34 games, level with Southampton and five behind 17th placed Swansea City who both have a game in hand, while West Ham are 14th with 35 points.
“If we can do that (beat Burnley), it will set the cat right among the pigeons,” said Lambert. “We’e playing well, we have an incredible support behind us, and I’m pretty sure come Sunday, there won’t be many stadiums rocking the way ours can rock.
“If the supporters do what I think they’ll do, we just have to turn up and try and get a win. Regardless of how we do it, we have to get a win but we certainly will drag teams into it.”
West Ham had three efforts ruled out after Marko Arnautovic was twice offside and Carroll handled inside the box but Lambert said his team were due a bit of luck and felt they deserved to come away from east London with a point.
He added: “I must have walked under a ladder or broken a mirror somewhere because I have not had any luck whatsoever.
“I was due it, the team was due it, the football club was due it and, if it is offside, it doesn’t matter if it is marginally offside, or close, it is offside.”
Moyes surprised at keeper Hart’s costly error
West Ham United manager David Moyes said he was surprised by goalkeeper Joe Hart’s mistake that handed Stoke
City the opening goal in Monday’s 1-1 Premier League draw but he defended the England player after his side had to settle for a point.
Hart was outstanding in the Hammers previous game against Chelsea and earned a man of the match display as his side came away with a crucial point in a draw at Stamford Bridge.
The West Ham keeper would have been desperate to impress once again ahead of the World Cup, with England manager Gareth Southgate watching from the stands, but he produced a major error which led to Stoke scoring after 79 minutes.
Hart had failed to hold a shot from Xherdan Shaqiri and substitute Peter Crouch was on hand to score from close-range.
But striker Andy Carroll came off the bench to spared his blushes with a superb equaliser in added time to salvage a draw and move the Hammers a step closer to safety.
West Ham have 35 points from 33 games and Moyes feels they need one more victory to guarantee survival and said he would not blame Hart for their failure to take all three points.
“He had two very similar shots in the first half and he took them quite comfortably in his midriff. So it was a surprise that it spilt out. But, look, if you’re a goalkeeper it’s going to happen,” Moyes told a post-match news conference.
“For the team, for us, at the time it was difficult because it meant we had to go and look to change after we had been the dominant team, apart from the first five or 10 minutes of the second half.
“Last week after he made fantastic saves at Chelsea, I said to him: ‘Well done’. I didn’t go over the top because I expect him to do that. Tonight I’ll tell him: ‘You should have done better on that one.’
“I think the points we’ve taken in recent weeks have been really important, winning here against Southampton (3-0) and taking a point at Chelsea (1-1). I think the important thing was to make sure we didn’t lose tonight.