The Cottagers had to ride their luck in the game, but Aleksander Mitrovic scored in injury time to give them all three points.
But after the game, manager Scott Parker said they were worthy winners, despite both sides having chances to open the scoring throughout the contest.
The win moves Fulham into third in the table, five points from automatic promotion and well in the playoff picture.
Parker told BBC Radio London: "We didn't get out of jail, I thought we were exceptional from start to finish.
"They came with a little bit of pressure but we moved the ball well, there was an attacking impetus about us and we had a lot of conviction.
"You miss a penalty and you think 'maybe it's not our night' but full credit to my team the commitment they showed, the belief, the mentality and in the end you deserve your moment to win the game."