The Dane has left the Swans with immediate effect on Tuesday night with veteran defender Garry Monk and first-team coach Alan Curtis to take charge of the team ahead of Saturday's clash with rivals Cardiff.
The news ends considerable speculation over Laudrup's position, which had been growing since last weekend's loss to West Ham.
The Capital One Cup-winning boss' relationship with the club had reportedly been strained since a major disagreement over transfer policy last summer.
The 49-year-old had also been under increasing pressure after a run of just one win in 10 Barclays Premier League matches left the Swans just two points above the relegation zone.
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 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, the former Real Madrid, Barcelona and Denmark midfielder, was appointed in the summer of 2012 having previously managed at Brondby, Getafe, Spartak Moscow and Real Mallorca.
He initially carried on the good work of Brendan Rodgers, who oversaw promotion and took the club to 11th in the Premier League before leaving for Liverpool.
Last season's Capital One Cup success - Swansea's first major trophy - enhanced Laudrup's reputation considerably and he was rewarded with a lucrative new contract until 2015.
However, problems surfaced last summer when it is thought Laudrup and Jenkins fell out over transfer targets. The difficulties led to the club's refusal to deal with Laudrup's representative Bayram Tutumlu in transfer deals.
Press Association Sport understands tensions have also remained throughout the current season and there has also been disquiet about the intensity of training sessions.
Prior to Laudrup's exit, it had been understood that Monk, who has not played since September after undergoing knee surgery, was being lined up for a coaching role.
The long-serving Monk, 35 next month, joined the Swans when they were in League Two in 2004 and he and Curtis have now been put in charge for the "foreseeable future".