The 36-year-old Points, who had struggled with seven missed cuts from nine events for the season, shrugged off the dismal form with nine birdies and a solitary bogey to be one stroke clear of countrymen Cameron Tringale and John Rollins.
A week after being unseated by Tiger Woods as world number one, Northern Ireland's McIlroy battled to a one-over 73 after a poor front nine saw him reach the turn at three-over.
McIlroy found some form with four birdies coming in, but added another two bogeys to be well off the pace in his last warm-up before Augusta.
"I felt like on the front nine I was a little tentative and I made a couple of silly mistakes, making double on the par-five there so I just went down and tried to commit to my shots more on the back nine and it seemed to help," McIlroy said.
"There were still a couple of loose shots here and there but all and all it was a tricky day and if I could have got it back to even par I would've been pretty happy.
"But I'm one-over par and hopefully conditions are a little calmer tomorrow morning and I can go out there and make some birdies."
Points, bidding for his second title after the 2011 Pebble Beach Pro-Am and a second start at the upcoming Masters, started on the back nine and birdied his first five holes out of the gate before another on the 17th for a scorching 30 at the turn.
Adding further birdies at three, seven and eight, Points rolled in over 160 feet of putts through the round with a new putter, but said he had been struggling mentally in the lead-up.
"It's super difficult (to stay positive). It's maybe the hardest thing I've ever had to do but I just keep plugging away and keep trying to do the right things," he told reporters.
"My ball striking was pretty good but my putting was a little unconscious ... I holed a few really long ones and I holed a bunker shot but all in all it was good golf.
"I really only made one bad swing and it was fun to finally see the ball go in the hole."
A number of players are battling to crack the world's top 50 after the tournament's conclusion and earn a last-minute invitation to the Masters.
Augusta native Charles Howell III and Swede Henrik Stenson shot 69 to be tied 18th, while 2006 US Open champion Geoff Ogilvy, Australia's last Major winner, was level with McIlroy at one-over.
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