The 14-time major champion, playing in his first full PGA Tour event for almost 18 months, carded a level-par second round of 72, but missed the cut by four shots after his first-round 76.
World number one Jason Day and US Open champion Dustin Johnson also missed the cut at Torrey Pines in San Diego.
Overnight leader Rose carded a one-under 71 to put him on eight under.
Canada's Adam Hadwin and USA's Brandt Snedeker are tied in second on seven under, while US PGA champion Jimmy Walker missed the cut as he finished on three over.
Woods is playing in just his second tournament since 15 months out with a back injury.
"It's frustrating not being able to have a chance to win the tournament," said the 41-year-old, who won his last major, the US Open, at the same course in 2008.
"Overall today was a lot better than yesterday. I hit it better, I putted well again. I hit a lot of beautiful putts that didn't go in, but I hit it much better today, which was nice."
Scotland's Martin Laird and England's Paul Casey are both on two under, while Ireland's Shane Lowry is on level par.

Jonathan Forte hit the post for the visitors, who are six points above the bottom four, before Tareiq Holmes-Dennis sent an effort wide.
Oldham keeper Joel Coleman saved a David Worrall shot at the other end.
Main fired home after the hour mark as he latched on to Forte's pass to seal victory for the Latics.
Oldham have a sizeable cushion above the League One relegation places with three games left to play, while Southend remain in 13th place.

Storms smashed a hole in the sea wall alongside the line above the seafront in Dawlish, south Devon in 2014.
The railway line took two months to repair at an estimated cost of £1bn to the local economy.
Network Rail is consulting with Devon and Cornwall residents about the plans.
They include strengthening defences and cliffs, as well and building breakwaters to protect the line.
More on the coastal rail protection work, and other Devon and Cornwall news
Cliffs next to the line would be stabilised and reshaped so they are not as steep under the plans.
A total of £270m is allocated for spending on priority areas including around Dawlish between 2019 and 2024.
The line connects Devon and Cornwall with the rest of the UK.
Tiger Woods missed the cut at the Farmers Insurance Open, as England's Justin Rose maintained a one-shot lead.
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Oldham Athletic boosted their hopes of League One survival as Curtis Main's second-half goal sealed a vital three points against Southend United.
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Protecting a key coastal railway track from rising sea levels and falling cliffs is going to cost about £650m, Network Rail has said.