The Potters had managed to claw back a two-goal deficit at Aston Villa last time but could not perform the same heroics against the Toffees, who had raced into a two goal first-half lead.
Stoke were poor in the first period but came out a different side in the second-half and pulled one back via a Ryan Shawcross header seven minutes after the re-start.
They then put Everton under severe pressure before Marouane Fellaini netted in injury time to seal the win for the Toffees.
The result saw the Potters drop into the relegation zone and Pulis was left to bemoan his side's poor start.
OffsideHe told Sky Sports that Stoke were disappointed with the second goal by Joleon Lescott - which was a tap in after Thomas Sorensen had saved a Tim Cahill header - as they thought it was offside.
"But he (Cahill) still got a free header we should deal with that ball that comes into the box," lamented Pulis, who re-iterated that Stoke thought it was offside but "sometimes it goes for you, sometimes it doesn't".
He added: "But the second-half was more like us, we really got to grips with the game and they scored late on with us really chasing the game. But for the majority of the second-half we took the game to Everton."
Asked if Stoke were giving themselves too much of a mountain to climb in games like this, he replied: "Yes, I think that we've done that loads of times where we really, really have come out of the blocks slow and we'll have a chat about it Monday.
"But the important thing is that we dust ourselves down - there are nine games to go and we have got to make sure now that we are right on it for the last nine games.
"I am pleased with the second-half but desperately disappointed again with the way we started - like I said, we will have to look at that because we have started slowly in too many games, especially away from home."