The Reds won both the Carabao Cup and FA Cup after beating Chelsea following a penalty shoot-out at Wembley on each occasion – but saw Manchester City finish a point ahead of them to claim the Premier League title on a dramatic final day of the campaign.
Liverpool will aim to add the Champions League trophy to their haul when they meet Real Madrid in Paris on Saturday.
Klopp also received the LMA’s Sir Alex Ferguson Trophy, which is voted for by the full membership of managers across all the divisions.
“It is a great honour and it was an insane season,” Klopp said. “The last matchday when only two games were meaningless and in the rest, we all played for absolutely everything.
“It was not the best outcome for us, but we are already over it.
“When you win a prize like this you are either a genius or you have the best coaching staff in the world – and I am here with all of my coaching staff, they know how much I appreciate them.”
Others on the LMA’s Premier League’s Manager of the Year shortlist were City boss Pep Guardiola, Thomas Frank, who kept Brentford up following a memorable first campaign, Eddie Howe at Newcastle and Crystal Palace boss Patrick Vieira.
Klopp had also won both of the LMA accolades in the 2019-20 campaign, which was interrupted by the Covid-19 pandemic.
“This (Sir Alex Ferguson Trophy) being voted for by my colleagues is obviously the most important prize you can get,” the German said.
“I don’t believe in individual prizes in football generally, it is a team sport and I would be nothing without these boys there. It is all about what we can do together and what we did together.
“We knew that we would be better this season, but had no idea we would be consistently better and I am really proud of the boys for what they did.
“We live in a world where second place will not be remembered – because of the other guys from Liverpool, you have to win all of the time – and Pep Guardiola deserved that obviously with Manchester City.
“Now we have the chance to play against the most experienced Champions League team and manager, but we were there now (in the final) in three of the last five years, so we will give it a try.”
The LMA Championship Manager of the Year went to Fulham’s Marco Silva, who guided the Cottagers straight back into the top flight, scoring 106 goals as they finished two points ahead of Bournemouth.
Wigan boss Leam Richardson won the League One award, with Matt Taylor named the League Two Manager of the Year having taken Exeter up, but just missing out on the title to Forest Green by goal difference.
Chelsea’s Emma Hayes was voted the FA Women’s Super League Manager of the Year, with Liverpool’s Matt Beard claiming the Women’s Championship award.
The LMA Service to Football Award went to Manchester United’s receptionist Kath Phipps, who has been working at the club for more than 50 years.
QPR’s Chris Ramsey and Manisha Tailor received the Kick It Out and Sky Inclusion Champion Award while Dr Sally Harris of HCA Healthcare UK was given the LMA Special Recognition Award.