The 15-time golf major champion, 45, had to be “extricated from the wreck” by firefighters and paramedics.
A statement said he had suffered “open fractures affecting the tibia and fibula bones” in his lower right leg and a rod had been inserted.
Injuries to his foot and ankle have been stabilised with screws and pins.
“He [Woods] is currently awake, responsive and recovering in his hospital room,” added the statement.
Woods ‘not able to stand’
Woods was “fortunate to be alive” said a deputy from the LA County Sheriff’s Department, who was first on the scene of the accident at 07:18 local time on the border of Rolling Hills Estates and Rancho Palos Verdes.
Carlos Gonzalez said the golfer was “not able to stand under his own power” before being removed from the vehicle which had sustained major damage.
He said that the American was wearing his seatbelt and was “still calm and lucid” when he told him his name.
“It’s very fortunate that Mr Woods was able to come out of this alive,” he added.
LA County Sheriff Alex Villanueva said that Woods’ car “crossed the centre-divider to the point that it rested several hundred feet away” and “that indicates they were going at a relatively greater speed than normal”.
He added: “However, because it is downhill, it slopes and curves, that area has a high frequency of accidents. It is not uncommon.”
He said the car hit a kerb, a tree and rolled over several times. Villanueva added “there was no evidence of impairment” with Woods.
Stars wish Woods well
Former US president Barack Obama, tennis great Serena Williams, basketball star Stephen Curry and singer Mariah Carey were among the many celebrities who posted their messages of support on social media.
World number three Justin Thomas is one of Woods’ best friends on the tour and expressed concern for his compatriot.
“I’m sick to my stomach,” an emotional Thomas said. “You know, it hurts to see one of your… closest friends get in an accident.
“Man, I just hope he’s all right. Just worry for his kids, you know. I’m sure they’re struggling.”
A statement from PGA Tour commissioner Jay Monahan said: “On behalf of the PGA Tour and our players, Tiger is in our prayers and will have our full support as he recovers.”
Woods’ personal struggles
Woods was involved in a car crash in November 2009 which eventually led to admissions of infidelity and the breakdown of his marriage. He then took a break from golf but returned shortly afterwards.
Following five wins in 2013, Woods started just 24 events in the next four years because of chronic back pain and multiple surgeries.
In 2017 Woods was arrested on suspicion of driving under the influence when he was found asleep at the wheel of his car. He later pleaded guilty to reckless driving.
He had five prescription drugs in his system as he recovered from the spinal fusion surgery that ultimately gave him a second golfing career.
Woods ended an 11-year wait for a major title when he won the Masters at Augusta in 2019.
He jointly holds the record for most PGA Tour wins at 82 with Sam Snead and is three behind Jack Nicklaus’ mark of 18 major titles.
Woods recently said he had hoped to play in this year’s Masters in April after having a fifth back surgery in January.
He ended 2019 sixth in golf’s world rankings but is now 50th.