Swansea began life after Laudrup on Wednesday as Garry Monk took charge of his first training session.
Long-serving defender Monk, now installed as head coach, worked with first-team coach Alan Curtis to put the Swansea players through their paces as preparations continued for Saturday's south Wales derby clash against Cardiff.
Swansea announced on Tuesday night the decision to part company with 49-year-old Laudrup, who had overseen a solitary win from the last 10 Barclays Premier League games.
Swansea said Monk and Curtis have been put in charge "for the foreseeable future".
The Trust, which has more than a 20 per cent shareholding of the football club, supported the decision and called on fans to get behind Monk and the players "in the tough battles ahead".
Trust chairman Phil Sumbler said the Trust had been aware of "growing concerns" among the Swansea City board regarding Premier League performances.
In a statement, the Trust said: "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, and ultimately we believe that this decision has been taken with the best interest of the club in mind."
And Sumbler added: "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 has also placed on record its thanks to Laudrup for his achievements during his time at the club.
Despite Swansea winning the Capital One Cup during Laudrup's reign, his relationship with the club had reportedly been strained since a major disagreement over transfer policy last summer.
Announcing Laudrup's departure, Swansea chairman Huw Jenkins said: "It is a decision we have taken reluctantly, but it's a decision made in the best interests of Swansea City Football Club and our supporters.
"It is the first time in nearly 10 years that the club has parted with a manager in this way, but we had to remove the constant uncertainty surrounding the club and Michael's long-term future with us.
"I had a meeting with Michael today (Tuesday) in a final attempt to support him and establish a way to improve the work of the backroom team to secure the results we need over the final 14 Premier League games.
"However, after thinking long and hard about the best way forward, I felt it was unlikely we would achieve a stable environment at the club to allow us to get back to basics and produce the performance levels that have served Swansea City so well over the last few years.
"Now we need to put that uncertainty behind us and move forward as a united football club on all fronts, while placing on record our gratitude to Michael for the work he has done over the last 18 months and wish him well for the future."
Laudrup, who had previously had managerial spells with four different clubs including Brondby, Real Mallorca and Getafe, was appointed in 2012 as replacement for Brendan Rodgers.
Problems first surfaced last summer when it is thought Laudrup and Jenkins fell out over transfer targets.
Press Association Sport understands tensions have also remained throughout the current season, and there has also been disquiet about the intensity of training sessions.
Swansea later confirmed on the club website that three of Laudrup's backroom staff have also departed the Liberty Stadium.
"The club can confirm that assistant manager Morten Wieghorst, fitness coach Oscar Garcia and overseas scout Erik Larsen have all been relieved of their duties," Swansea said.
"The club would like to thank them for their services."
Laudrup handed former Denmark, Celtic and Dundee midfielder Wieghorst the assistant manager's role a year ago.
Wieghorst, 42, was Denmark Under-21s manager before moving to Wales.