Principal owner John W Henry, Liverpool chairman Tom Werner and FSG president Mike Gordon also thanked fans "for telling us when we've fallen short" in a message on the club's website.
Under the Americans' ownership, Liverpool won the English league title for the first time in 30 years in 2020, adding another Premier League crown last season, and celebrated a sixth Champions League triumph in 2019.
It has not all been plain sailing, with significant missteps including the club's ill-advised involvement in the proposed European Super League, which prompted a swift apology and U-turn from Henry.
Club bosses also scrapped controversial ticket hikes after a fan protest.
"Off the field, there have been times we've got things wrong," the management trio said in their joint statement.
"We know that and we have learned from it. All our decisions are made with the best, long-term interests of the club at the centre of our thinking."
Addressing the fans, they said: "Thank you for welcoming us, for believing in this team through every high and low, for telling us when we've fallen short and for showing the world what it truly means to be part of Liverpool Football Club.
"Today is a day to look back with gratitude. But it's also a reminder that our work isn't done. There's still so much more ahead of us - more to achieve, more to win, and more memories to make, together."
Under FSG, there has been significant investment in infrastructure at Anfield and a new training ground has been constructed.
But their most visible achievements have been on the pitch, celebrating a record-equalling 20th English league title last season.
"From the outset, our primary aims were simple: to bring success back to Anfield and to ensure the long-term health and stability of this great club," said the owners.
"We understood the responsibility that came with being its custodians, and we've tried to honour that every single day."