After Potters defender Robert Huth had headed wide on the quarter-hour mark, Charlie Adam put the hosts in front in the 26th minute with a shot on the turn from close range.
Stoke continued to carve out opportunities, with captain Ryan Shawcross going closest by nodding against the bar, but were unable to add to their tally.
That made for some anxious moments for their manager, particularly towards the end as Fulham stepped up their search for an equaliser, and reflecting on the contest, Pulis said: "I'm really disappointed we have not won the game by more goals.
"Even just looking at the first half, if you look at the chances, Robert has missed a point-blank header from a corner and Ryan hits the bar when he is almost under the goal.
"There were some great opportunities and in the second half we had more.
"Then in the last few minutes, you are starting to scramble.
"We had so many chances and you get a little bit uptight because you know one mistake could have cost us the game - a game that we should have been out of sight in."
Adam's goal - his second as a Stoke player - did ultimately prove the difference, though, just like in the previous home match when the Scotland midfielder opened his Potters account to secure a 1-0 victory over QPR.
Pulis has made it clear since signing Adam from Liverpool in August that he is looking for goals from him and he was delighted to see the 26-year-old in the right position this afternoon to make the most of a knockdown from striker Peter Crouch.
"We have just asked Charlie to follow Peter around - to get closer to Crouchy, because we think he has real ability and we want him to do it in the final third of the pitch rather than the other two thirds," Pulis said.
"For a while he has just been Charlie, playing his own game. He is like a schoolboy at times - he just goes out there and loves the game of football.
"But we are pretty structured, organised and disciplined, and we need him up around Peter.
"If he gets up there enough, Crouchy does fantastically for us and Charlie will get the opportunities to score some goals this season."
The result, which extends Stoke's unbeaten Premier League home record to 13 games stretching back to February, sees the Potters move up to 11th in table, level on 16 points with ninth-placed Fulham.
Cottagers boss Martin Jol admitted his side had struggled, particularly early on, to get into the contest.
"In the second half we did better - in the first half it was exactly the same as it is for most of the teams here, and probably how it was for us last year," Jol said.
"We couldn't get out of our half. We still wanted to play, but it was difficult and then you get the long balls and they close you down, pressurise you, play their style.
"It was not easy for us - we play a different style."