Matt Lowton hit an unstoppable long-range volley three minutes from time at the Britannia Stadium to put relegation-threatened Villa back ahead in a tense Barclays Premier League encounter.
Michael Kightly had cancelled out Gabriel Agbonlahor's early goal just minutes earlier but Lowton's effort was followed by a late Christian Benteke goal as Villa claimed a deserved 3-1 success.
The result was greeted by huge relief by Villa, who began the game in the bottom three, but it also pulled Stoke into the scrap after a run of one win in 13 games.
Pulis' side may have been in a respectable-looking 13th place ahead of the game but the loss left them just a point above Villa.
Pulis said: "I think when things go against you they go against you and at the moment things are going against us.
"In life when things are going well and everyone is happy it all rolls and doesn't take any character to roll with the good times.
"When the bad times come you have got to show your character and that is what we have got to do as a group - everyone at the football club. Everyone has got to stick together and make sure we get the six points to stay up."
Pulis brushed off suggestions he could be fighting for his job. He added: "That is up to (chairman) Peter and the Coates family. I have always given my best for the club and always will. Hopefully, fingers crossed, we will get enough points to stay in the Premier League.
"We want 40 points. That is all we want and nobody has every heard me say we want this, that or the other. The first target is 40 points and I never get blase about it. Other people might, but I don't."