Stoke had dominated for large periods but spurned a gilt-edged opportunity to double their lead before the break when Mame Biram Diouf miscontrolled the ball in the box.
Lambert's charges are now four points adrift of safety with just three games to go but, despite being winless in 11 matches, the Scot retains hope of beating the drop.
"We have to win," Lambert told reporters at full time. "Until somebody tells us mathematically we can't do it, we won't give up. That's important as a team, that we never give up.
"We got off to a perfect start, looked great and obviously the big moment was Mame's chance, which was huge for us, but I cannot fault Mame Diouf for commitment and giving us everything.
"I think he needed that [arm round the shoulders]. I would never criticise any player if they gave me everything and Mame has certainly done that since I came here."
Stoke's final three games of the season will see them travel to Liverpool and Swansea City, either side of a home game against Crystal Palace.