Badou Ndiaye's early goal had given the Potters the lead their first half performance deserved, but Burnley came out fighting after the break and got an equaliser thanks to the in-form Ashley Barnes.
The turning point of match came shortly before half time when Moritz Bauer's cross found Mame Biram Diouf, who allowed the chance to get away from him when he should have made it 2-0 from close range.
“We made the perfect start, scored early and really looked to be in the ascendancy," Lambert told Stoke's official website. “We had a wonderful opportunity to really kill the game off just before half time when the ball fell to Mame, but he was perhaps a little too hesitant.
“Had he pulled the trigger, and the ball had hit the net, then I don't think there would have been a way back for them."
The result means Stoke will probably need to win at least two of their remaining three games, and even that might not be enough to save them from the drop.
“For as long as it is mathematically possible then we will keep fighting with everything we have," added Lambert.
“We wanted the win today, we didn't get it but we got a point, and in football who knows - it may prove to be a hugely important one.”
Stoke are bidding to extend their stay in the Premier League to 11 seasons, having been promoted in 2008.