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.
Potters boss Pulis had been to Istanbul on Sunday to watch Besiktas play and knew they were always going to pose a threat.
"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."
One of Stoke's substitutions in the second half was enforced, with Matthew Etherington having to come off for Jermaine Pennant shortly after the restart.
Etherington's condition will be assessed ahead of the Barclays Premier League trip to Swansea on Sunday.
"Matty has got a little strain, so we will have to wait to see what he is like," Pulis said.
Besiktas coach Carlos Carvalhal was unhappy with referee Antony Gautier's decision to award the spot-kick and also thought his side should have had a penalty of their own.
"Stoke won, but they were not the best team on the field - in my opinion, that was my team," Carvalhal said.
"If you give a penalty in that situation, in these matches there will be 10 or 20 penalties, because players come together all the time.
"The officials also didn't see a handball in the area, and we are not comfortable with this."
As far as Pulis was concerned, though, there was no debate to be had about the incident that led to the Potters' winner.
He said: "I haven't seen it, but it is a penalty because the referee gave it."
Source: Team Talk
Source: Team Talk