The 62 would have matched the course record if not for the preferred lies rules in play.
Points shares the lead at 8-under-par 205 with Matt Bettencourt, who posted a 5-under 66 in round three.
Gavin Coles (68) and Josh Broadaway (69) are tied for third place at minus-7 with second-round co-leader Marc Leishman (70).
Points, who started the day tied for 43rd, played the back nine first. He opened with a birdie on the 10th, then caught fire at the 12th.
He drained a birdie effort on 12 and came right back to birdie No. 13. Points kept rolling with birdies on 14 and 15 at Miccosukee Golf & Country Club.
Points, who is 40th on the money list, made it five straight as he birdied the par-5 16th from 22 feet out. However, he gave that shot right back with a bogey on 17. Around the turn, he birdied the second to get back to minus-5.
The three-time winner on the Nationwide Tour made eagle on the par-5 fifth from about 35 feet. Points rolled in a 4-foot birdie putt on the ninth, his last, to finish at 8 under.
'I tried really hard to stay as laid back as possible and it translated into making some putts,' said Points. 'I knew if I could get it under par for the week, anything can happen. To make some putts the last couple of days has been really exciting.'
Bettencourt posted birdies on the fifth and seventh to get to minus-5. After tripping to a bogey on nine, he connected on back-to-back birdie efforts from the 10th.
The 33-year-old, who won the Oregon Classic earlier this year, birdied the par-5 16th and made it two in a row with an 18-foot birdie putt on 17. He parred the last to share the third-round lead.
'I've been putting really well and that makes up for a lot of mistakes,' said Bettencourt, who stands 22nd on the money list. 'I'll stay focused tomorrow, try to put together a good round and whatever happens, happens. Hopefully, it's good enough to get a victory, but if not I'd like to finish in the top 5. That would secure my spot for next year.'
Daniel Summerhays carded a 4-under 67 to move into a share of sixth at 6-under 207. He was joined there by Scott Gardiner (68) and Bryce Molder (69).
Chris Smith and Reid Edstrom are one stroke back at minus-5.
Ricky Barnes, who shared the second-round lead with Leishman, faltered to a 3-over 74. That dropped him into a tie for 15th at minus-3.
There was a 25-minute weather delay, but the round completed before further storms hit the course.