The group paid tribute to the Dane for his achievements and acknowledged a mixed fan reaction to the decision but believe the board have the club's best interests at heart
The club made the announcement on Tuesday, with Garry Monk taking over as new head coach "for the foreseeable future".
Swansea had won just once in their past 10 Premier League matches and the trust, which has more than a 20 per cent share in the club, stated that they were fully behind the board.
"During its tenure at the club, the current board, including a representative of the supporters' trust, has made key decisions which have taken the club forward," said Trust chairman Phil Sumbler.
"Ultimately we believe that this decision has been taken with the best interest of the club in mind.
"There has inevitably been a mixed reaction among fans to the announcement, with the club facing a critical period of key matches in three major competitions.
"However, we know from our supporter director Huw Cooze that the board has not taken this decision lightly, particularly in such an important week for the club, and it's vital that we now all pull together to give the team our full support.
"We all want to see an upturn in results and what better place to start than at the Liberty Stadium against Cardiff City."
The trust also went on to place on record their thanks to Laudrup, who led the club to their first major domestic trophy in winning the League Cup at Wembley in February 2013.