Officials were forced to apply a split-tee format with three-man groups in a desperate bid to complete the third round after 39 players returned early on Saturday morning to finish the weather-interrupted second round.
Playing possibly his last-ever Masters, 47-year-old Woods battled valiantly, walking the hilly course on his rebuilt leg, to make the cut on the number at two over par with a steely second-round 73.
The five-times champion joined Gary Player (1959-82) and Fred Couples (1983-2007) in the record books with the most consecutive Masters cuts made.
"I've always loved this golf course, and I love playing this event. Obviously, I've missed a couple with some injuries, but I've always wanted to play here," Woods said before the cut was confirmed.
"I hope I get a chance to play this weekend. I wish I get a chance to play two more rounds."
Couples, 63, also penned another piece of history by becoming the oldest man to make the cut at Augusta.
Thirty-one years after his 1992 victory, Couples carded a 74 to be one over and 13 shots behind leader Koepka.
"That's why I come here," said Couples, who last made the cut here in 2018. "The last four years have been really mediocre golf. Maybe one year I was semi-close to making the cut. But that's my objective, and I did it.
"Am I going to look thrilled to play 18 holes in this this afternoon? No, I'm a wimp. I'm an old wimp, but I'm excited to play. I love this place."
Realistically, Woods and Couples are making up the numbers with Koepka, at 12 under, Rahm, 10 under, and US amateur Sam Bennett, eight under, clear of the field.
Australians Jason Day, at five under, Cameron Smith and 2013 champion Adam Scott, both two under, are among those playing catch-up in the brutal conditions.
"With the temperature we had and how long the golf course was playing, I was very happy to be able to play the back nine under par," Rahm said after adding a 69 to his opening 65.
Temperatures dropped to eight degrees Celsius and left the remaining 54 competitors facing a gruelling test over the final two days in pursuit of the green jacket.
Australian pair Min Woo Lee and amateur Harrison Crowe both missed the cut.
Lee finished at six over after a second-round 75, with Crowe completing his Masters debut at eight over after a 77.
