The 24-year-old, who won two Eredivisie titles under new United boss Erik ten Hag in Amsterdam, has signed a contract until 2027 at Old Trafford, with the option of a further 12 months.
Ajax said they had agreed a transfer fee of 57.37 million euros ($58.2 million, £48.3 million) with United.
Including variables, that amount could increase to 67.37 million euros.
“It’s an honour to join this great football club,” the centre-back told United’s website.
“I’ve worked so hard to get to this moment and, now that I’m here, I’m going to push myself even further.
“I’ve been lucky enough to be part of successful teams in my career and that’s what I want to continue at Manchester United.
“There will be a lot of work to get to that moment, but I firmly believe that, under this manager and coaches, and together with my new team-mates, we can do it.”
Martinez, who began his career at Newell’s Old Boys and also played for Argentine side Defensa y Justicia, won the 2021 Copa America with his country and has been capped seven times at international level.
He is the third signing of the Ten Hag era following the capture of left-back Tyrell Malacia from Feyenoord and the free transfer of former Brentford and Tottenham midfielder Christian Eriksen.
Manchester United football director John Murtough said: “Lisandro is an outstanding player who will bring further quality and experience to Erik’s squad.
“We are delighted that he has chosen to join Manchester United and we are looking forward to seeing him develop further and help the team achieve the success we are aiming for.”
United, who finished a disappointing sixth in the Premier League last season, host Brighton on August 7 in their first match of the new campaign.