The Turkish visitors went ahead just before the quarter-hour mark when Roberto Hilbert struck into the far corner having latched on to Ricardo Quaresma's pass, but their lead lasted only a minute, goalkeeper Rustu Recber unable to claim Dean Whitehead's corner and Peter Crouch on hand to bundle the ball over the line.
The two sides, roared on by their equally-vociferous sets of fans, played out an entertaining contest from there, with both hitting the woodwork.
Substitute Jon Walters then settled things in the 78th minute, hammering a penalty past Rustu after Crouch had been brought down by Tomas Sivok, to secure all three points for Stoke and send them top of Group E.
"They are a very, very good side - I said it before the game," Pulis said.
"They have some good players, and some of them that played today had been rested on Sunday.
"We think they are a massive danger in this group, so we are very pleased with the result.
"I thought in the first half they played very well, outnumbered us in midfield and moved the ball very quickly. At times we were chasing the ball.
"But in the second half we made some changes, pushed on a bit and caused them a few more problems.
"On another day we could have scored three or four goals."