Deflected second-half efforts from Jonjo Shelvey and Chico Flores sealed a 2-0 victory for the Swans.
Fulham failed to make the most of excellent first-half chances for Hangeland and Ashkan Dejagah as they slipped to a defeat which ensures they remain just a point above the drop zone.
Laudrup had warned his team they may have to suffer in order to end their worrying winless streak, their worst run of league form since the 2008-09 season, and so it proved.
There was plenty of anxiety among the home support during a tight first hour, but the Welsh club have opened up some valuable breathing space ahead of the weekend trip to West Ham.
"I am relieved," said the Dane.
"It is a huge win. It is quite some games since we won in the league and it was against a direct opponent in the table.
"We have four in a row - with West Ham, Cardiff and Stoke to come - so they are all six-pointers.
"We will go into the game on Saturday with two consecutive wins - although only one in the league - but that will give the players the belief that we are going forward."
Laudrup and Swansea have been hindered by a lengthy injury list in recent weeks, but the Dane was able to welcome back Jonathan de Guzman, Shelvey, Nathan Dyer and Angel Rangel, with the likes of Michel Vorm, Michu and Jose Canas making good progress in their recoveries.
But Laudrup was keen to stress claiming three points was down to more than having more players available.
"Overall I am pleased that some injured players came back but I don't think it would be fair after coming through these games to talk about one or two players.
"When you are struggling mentally it is only the team together that can bring you out of that situation.
"If you have six or seven who are very tense then it is very difficult to play football.
"It was better today compared to the first-half against Birmingham and then we were very good after the first goal.
"The first goal in games like this is very crucial.
"We played too much short passing in the first half but it was not easy as Fulham came here to take away a point defending with nine players behind the ball."
Swansea have added David Ngog and Adam King to their squad ahead of the end of the January transfer window.
But the club are still waiting for Blackpool's Tom Ince to decide whether he wants to move to the Liberty Stadium on loan.
"I have already said he is looking at his options," said Laudrup.
"I talked to him and his father (on Monday) and I saw on the news they have been with another club, and they will make up their minds as to what they think is best for him."