On a rainy, windy, chilly day at the Muirfield Village Golf Club, Sabbatini shot a 33 on the back nine for his second straight 69 and his 6-under 138 total gave him a one-shot advantage.
Woods, seeking his second win of the year, shared second place with Scott Stallings and Spencer Levine. Daniel Summerhays and Jim Furyk were tied for fifth at 140.
There were 22 players within four shots of the lead with two rounds to go, but some of the game's current stars found the conditions too tough to survive.
Rory McIlroy, the world's second-ranked player behind Luke Donald, shot a 79 for a 150 total and missed the cut in his third straight event. He failed to survive to the weekend at the Players Championship and the European PGA Championship.
Current PGA champion Keegan Bradley missed the cut, as did Masters winner Bubba Watson.
Rain delayed play for almost 2 hours early Friday and the storms brought a cool and strong wind during the rest of the day.
Sabbatini, playing late in the afternoon, was only 2-under for the tournament heading to the ninth tee, but he played the final 10 holes in 4-under. The most recent of his six PGA Tour victories came last year at the Honda Classic.
Woods made a run at the lead despite recording his second double bogey of the week while putting together a round of 69. He needed five strokes to play the par-3 12th, the same hole on which McIlroy made a quadruple bogey Thursday.
McIlroy made a par on that hole Friday, but he suffered two double bogeys and three bogeys while failing to make a birdie. He is struggling through his worst slump since winning last year's U.S. Open, a title he will defend in two weeks at the Olympic Club in San Francisco.