In a major setback for a club that added star big man Anthony Davis to four-time NBA Most Valuable Player LeBron James in July, Cousins suffered the injury working out in Las Vegas on Monday when he "bumped knees" with another player, according to ESPN.
Team doctors in Los Angeles on Thursday confirmed the third serious leg injury for Cousins in 18 months, with no timetable set for surgery to repair the torn anterior cruciate ligament.
Cousins was signed as a free agent in July, on a one-year deal for $3.5 million, after playing with the Golden State Warriors last season.
He spent most of the past campaign recovering from a left Achilles tendon tear in January 2018 but played 30 games for Golden State before suffering a torn quadriceps in the second game of the playoffs that sidelined him until the opening game of the NBA Finals, which the Warriors lost to the Toronto Raptors.
Cousins, 29, has played nine NBA seasons, averaging 21.2 points and 10.9 rebounds a game.
Losing Cousins would force some key changes into the Lakers' frontline, potentially including moving Davis from power forward to center to replace Cousins.
It also figures to give backup center JaVale McGee more time in the middle, potentially a major role if the Lakers elect to keep Davis at forward, with playmaker James another versatile threat.
"It's super unfortunate for a guy like DeMarcus because you can just tell how much he loves basketball through all of these injuries," Lakers standout Kyle Kuzma said at a United States national team workout in Los Angeles on Thursday. "He was going to be a big part of what we're going to do."