The Toffees, who had had penalty appeals turned down early on when Glenn Whelan appeared to make a block with his arm in the box, took the lead in the 36th minute through Ryan Shawcross' own-goal, the centre-back inadvertently nodding in a Steven Pienaar cross.
Leon Osman wasted a clear opportunity to make it 2-0 just before the interval by blasting wide and the Potters capitalised after the restart with Kenwyne Jones heading home an equaliser in the 52nd minute, his first top-flight goal for 16 months.
Pulis said: "I thought it was a very even first half.
"I haven't seen it, but they tell me that Everton were unlucky not to get a penalty from what might have been a handball by Glenn.
"Then, I thought the big incident in the game was when Osman missed his opportunity - that was the changing point of the game for me.
"If we had gone in 2-0 down at half-time, it would have been very difficult for us to get back.
"I spoke to them at half-time and said 'we have been let out of jail with that one - let's push up an extra 10 or 15 yards and see what they are like when they have to defend properly'.
"Then I thought we created four or five great opportunities in the second half.
"We are disappointed in the end not to have won the game."