Stoke made it back-to-back league wins, sandwiched between a Europa League draw at home to Dynamo Kiev.
It was also only their second league victory on the back of a European match.
"This was a real battling performance against a good Everton team," said Pulis.
"It has been a great week for the club. To get six points and to qualify for the latter stages of the Europa League is first class.
"We have had a dip of form, which most clubs do, and we have shown a lot of character and spirit the last week and the result on Thursday against Kiev was a fantastic result for us."