The American former world number one, seeking his first PGA Tour victory in over two years, lost the first two holes to his opponent and trailed by one after 14 in a wildly fluctuating encounter before taking control.
Champion at this event in 2003, 2004 and 2008, Woods won the 15th with a birdie and 16th with a par before sealing victory with an eight-foot par putt at the 18th where he was bunkered with his approach.
"I don't think either one of us had our best stuff today, subsequently the match was back and forth," Woods told reporters at Dove Mountain's Ritz-Carlton Golf Club. "It was the epitome of match play.
"We both made our share of mistakes, there's no doubt about that, but somehow I was able to move on."
Woods, who will face fellow American Nick Watney in Thursday's second round, said he had struggled on the huge, severely contoured greens in dazzling sunshine at Dove Mountain.
"I hit a couple of bad putts, I'm not going to deny that," the 14-times major champion added. "But I had a hard time reading these greens.
"I was lucky to have had that putt on 18 where I saw his putt (first). His putt hung on the edge. You would think that my putt should be a left-to-right putt, it was actually right-to-left.
"So just trust it and see what happens. And I poured it in there."
SIZZLING START
Fernandez-Castano, who had described the 14-times major champion as "beatable" during the build-up to this week's event, made a sizzling start by sinking putts from 13 and 22 feet to birdie the first two holes.
His birdie at the par-five second was astonishing as he hit his drive well right into a small bush before striking a superb second shot to just short of the green and then getting up and down.
However, the Spaniard bogeyed the par-four fifth after missing the fairway to the right for his lead to be cut to one up before Woods wrested back the initiative with birdies at the seventh and eighth.
One up at the turn, Woods lost the 10th after pulling his drive left into desert scrub and also the par-five 11th, where Fernandez-Castano hit his third shot to three feet, to trail by one.
"It was up and down for both of us," said Woods. "He's two up, I have a chance to go two up, then I'm almost two down again. And then I look like I probably should go two down on 14 green. Then all of a sudden a couple holes later I'm one up."
The former world number one squared the match with a birdie at the driveable par-four 15th and edged one up at the par-three 16th where his opponent missed the green badly to the left.
Woods, who is recovering from a cold, had a golden opportunity to end the match on the 17th green but he missed a birdie putt there from 10 feet before getting up and down from the greenside bunker at the last to clinch the win.
"If there was one day to beat Tiger Woods, this was it," Fernandez-Castano said. "I didn't take the opportunity. I missed a few shots, and you can't miss spots if you want to beat one of the greatest in history."
