Mason Mount scored Chelsea's only goal of the game after 78 minutes of sustained pressure from Frank Lampard's side.
Robinson was sent off for a lunging tackle on Cesar Azpilicueta right before half-time, making him the eighth ever U.S. international to receive a red card in the Premier League.
It was an uneventful first half up until Robinson was sent off with Mount coming the closest to scoring when he hit the crossbar on 24 minutes.
Despite Chelsea's dominance in possession, they struggled to create chances as Fulham pulled every player into defence.
It was Mount who managed to break the deadlock after Ben Chilwell's shot was deflected right into his path.
"It was a massive game for us. We know the run we have been through," Mount said after the game.
"It has been tough. We have had to dig deep as a team.
"We all knew that we needed a win, we put that pressure on ourselves. There is definitely pressure on us to perform.
"We stayed patient. We tried to play. Them having a man sent off helped us as well."
Lampard was relieved after his side ended a three-match winless league run.
"Of course it feels good to get a win, it has not been easy with the league form recently and you have to break the duck," Lampard told Sky Sports at Craven Cottage.
"Just because of the current form it is a bit harder, the confidence is not quite there. But we found it so I'm really pleased.
"I trust the quality of the players. There was a feeling we would get a goal, because the pressure was constant. It was about staying patient with our passing and I think we did."
Lampard pointed out Chelsea struggled against a packed Fulham defence after Robinson was sent off in the 44th minute and urged Mount to venture forward more in order to get into scoring positions.
"It is a pressure to move the ball, hit the sides, open it up, make them run and create chances. We were doing that," said Lampard.
"Mason was in a deeper role but can still arrive. He needs to score more goals, he knows that, but his quality and attitude is brilliant."
The win takes Chelsea up to seventh place and three points off the top four while Fulham struggle in 18th.