Griffin, who started out on Friday three shots back of first-round leader Chad Ramey, shot a one-under-par 71 that brought him to six under on the week and gave him a two-shot clubhouse lead over fellow American Will Gordon (67).
South African Christiaan Bezuidenhout (28), four-under through 14 holes, and Canada's Adam Svensson (29), four-under through 11 holes, were the joint on-course leaders at eight-under on the week.
The only hiccup of the day for world number 78 Griffin came at the par-four 18th where he hit a wayward drive and struck a tree with his second shot en route to a closing double-bogey.
"Yeah, it's playing tough out there. To be bogey free through 17 holes I thought was pretty freaking good," Griffin told reporters.
"It was a grind, and unfortunately I didn't finish the way I wanted to, but the 18th hole is probably the toughest hole all day. I'm not going to be the first guy to make a double there. You're going to see a few."
Ramey was clinging to the lead in conditions made tougher by whipping winds until a disastrous quadruple-bogey at his eighth hole, the par-three 17th, where his first two shots settled in the water instead of safely on the island green.
Lurking three shots back of the on-course leaders is Masters champion and world number two Scottie Scheffler (26), who offset an early bogey with a pair of birdies and was one under on the day through 10 holes.
Scheffler, making only his third start at The Players, matched his low round at the event with an opening 68 and looked ready to make a move as he had just made two consecutive birdies around the turn when play was suspended.
"I wasn't swinging my best, but I made two nice birdies toward the end of the round and then obviously the delay will hurt a little, but start back up in the morning," said Scheffler.
Rahm withdraws
World number one Jon Rahm (28), the pre-tournament favourite who was seeking his fourth win of the season, withdrew from the PGA Tour's flagship event due to illness ahead of his second round.
With the projected cut line around one over, world number three McIlroy headlined the big-name players who were at risk of missing out on the final two rounds.
McIlroy, who began the day at four over par, needed something special but fell further back after he followed a bogey at the fifth with a double-bogey at the par-four sixth where he putted off the front of the green with his third shot.
The Northern Irishman responded immediately with a birdie and appeared to steady the ship with three consecutive pars but would need to catch fire over his final eight holes to have a chance at making the cut.
It was a whirlwind round for Jordan Spieth (29) who ground it out all day and arrived at his final hole, the par-four ninth, where he needed a birdie to ensure he made the cut.
Spieth's tee shot was headed toward the water when it hit a spectator's knee and bounced back into the fairway from where the three-times major winner sent his approach shot right of the green and then chipped in for eagle.
"I got an extremely lucky break on nine or I wouldn't be playing the weekend," said Spieth.
"Trying to get that guy's information and see literally whatever he wants this weekend because everything from here on out is because it hit him."
Among the other players in danger of missing the cut are reigning PGA Championship winner Justin Thomas (29) and 2022 US Open winner Matt Fitzpatrick (28).
