Woods, who defeated David Toms in last year's final, trailed throughout the entire match but took advantage of mistakes made by his 16th-seeded opponent.
Woods found himself 2-down after Rollins cashed in a short birdie putt at the 12th.
But Woods rebounded with a win at the 13th and a bogey by Rollins at 17 knotted up the match.
At the par-5 closing hole at La Costa, Woods drove into the right rough but was able to lay up down the fairway. Rollins found the fairway off the tee and then again with his second but that's when trouble set in for the 2002 winner of the Canadian Open.
Rollins pushed a pitching-wedge into the bunker on the right side of the green while Woods hit his third 20 feet left of the hole. Rollins' bunker shot never even made it to the putting surface and his fifth ran four feet past the hole, meaning Woods had two putts to win the match.
Woods didn't need two putts as his birdie try fell right into the cup.
Despite surviving the near upset, Woods was not happy with the state of his game or his performance on Wednesday.
'I kept making mistakes,' said Woods, who lost to Peter O'Malley in the first round two years ago when he was the No. 1 seed. 'I made three bogeys today which is unacceptable.'
Rollins may have let a gigantic opportunity slide by but was relatively upbeat after the match.
'I did what I wanted to,' said Rollins. 'I gave him a match. I'll probably be a little disappointed. I feel like I missed a couple good opportunities. I'll take a lot from it.'
Woods will now battle Trevor Immelman, the No. 8 seed in the Bobby Jones bracket, in Thursday's second round. Immelman moved on thanks to a 2-and-1 victory over Shigeki Maruyama.
The three other No. 1 seeds moved on to the second round.
Vijay Singh, the top seed in the Gary Player bracket, trounced Shingo Katayama, 5 and 3. Mike Weir, last week's Nissan Open champion and No. 1 seed in the Ben Hogan bracket, beat 2002 PGA Champion Rich Beem, 3 and 2, and Davis Love III, tops in the Sam Snead bracket, defeated Briny Baird, 2-up.
Love now gets a match with good friend Fred Couples in the second round after Couples bested Bob Tway, 3 and 2, in the first round.
The biggest upset in Wednesday's first round involved Retief Goosen, the No. 2 seed on the Hogan side. He was defeated by John Huston, 2 and 1, and was the highest seed eliminated in the first round.
Phil Mickelson continued his strong form in 2004 as he played well in beating Lee Westwood 3 and 1.
Both of last year's stunning major winners advanced on Wednesday. British Open winner Ben Curtis defeated Charles Howell III, 2-up, and PGA Champion Shaun Micheel needed 21 holes to oust Englishman Paul Casey, 1-up.
Fifteen matches went the full 18, while six needed extra holes to crown a winner.
Darren Clarke and Eduardo Romero nearly matched the tournament record for longest non-championship match, but Clarke won the 25th hole to earn the 1-up victory. Weir and Loren Roberts went 26 holes last year.
'It was very tough,' said Clarke, who holed a chip for eagle at 18 to extend the match. 'Eddie had a couple of really good chances early on in the playoff holes. Unfortunately he didn't quite make them.'
Clarke will now face Alex Cejka in the second round after Cejka beat Justin Leonard, 4 and 3.
The remaining winners in the Player bracket were: Kenny Perry (6 and 4 over Jeff Sluman), Steve Flesch (1-up in 19 holes over Brad Faxon), Chad Campbell (3 and 2 over Tim Herron, Roberts (1-up over Jay Haas) and Jerry Kelly (1-up over Sergio Garcia).
Woods and Immelman will be joined in the second round of the Jones bracket by: Padraig Harrington (2 and 1 over Toshi Izawa), Bob Estes (1-up in 19 holes over Scott Verplank), Toms (1-up in 19 holes over Niclas Fasth), Micheel, Thomas Bjorn (4 and 3 over Scott Hoch) and Fredrik Jacobson (5 and 4 over Phillip Price).
Weir and Huston are in the second round of the Hogan bracket and also in are: Duffy Waldorf (5 and 4 over Jonathan Kaye), Ian Poulter (1-up over Chris Riley), Peter Lonard (1-up over Rocco Mediate), Colin Montgomerie (1-up in 20 holes over Nick Price), Stewart Cink (4 and 2 over K.J. Choi) and Stephen Leaney (1-up over Fred Funk).
In addition to the Love/Couples match and the Mickelson/Curtis match, the Snead bracket will feature Stuart Appleby (5 and 4 over Justin Rose) against Chris DiMarco (2-up over Michael Campbell) and Robert Allenby (3 and 2 over Brian Davis) versus Adam Scott (2 and 1 over Miguel Angel Jimenez).