Carlos Franco and Jim Furyk, who owned the lead briefly, are tied for third at 13-under-par 203. Canadian Mike Weir is one shot further back at minus-12.
Hoch started slowly, but rebounded with seven birdies over his final 11 holes. The 10-time winner on tour bogeyed the par-5 first, posted six straight pars, then carded three consecutive birdies from the eighth.
'After five holes, I told my caddie, 'It doesn't look like we're going to get it done today.' I was trying different swings and it wasn't working,' said Hoch. 'All of a sudden, things clicked on No. 8 and I started hitting good shot after good shot.'
The 2001 Advil Western Open winner kept rolling along with another birdie at the par-5 12th on the Blue Course of the Doral Golf Resort and Spa.
Hoch later notched back-to-back birdies from the 15th to move to 13-under. He birdied the par-4 18th to claim a share of the lead.
Hoch is hoping that his wrist doesn't start to bother him like it did when he won the Western Open.
'I know the Western a few years ago when I won, I had to quit during my practice round because my wrist was bothering me so much after about 12 holes,' said Hoch. 'That was kind of a thing that it would bother me a good bit but I know if it came on, it would be best if I laid off. So I took it easy the rest of the day and I probably played in the pro-am and ended up winning the tournament.'
Tway, who owned a one-stroke lead heading into the third round, opened his round with a two-putt birdie at the first. He later birdied the par-3 fourth from 20 feet out, but gave that shot back with a bogey at the next.
The 1986 PGA Championship winner two-putted for birdie at the par-5 eighth to make the turn at 13-under.
On the back nine, Tway birdied the 10th to move to minus-14. He then parred the final eight holes to hold onto a share of the lead.
'I did some good things today and some things I need to improve on, but overall it was a fairly solid round,' said Tway. 'I was able to take advantage of the first three par-5s. I was able to hit those in two and two-putt for birdies and only made one bogey, that's always good. I need to hit the irons closer to the hole to give myself more opportunities.'
Franco started the day two shots behind Tway, but was able to chip away at that deficit. He collected a birdie at the first and later notched birdies at Nos. 6 and 8.
The three-time PGA Tour winner faltered with a bogey at the par-3 ninth. He erased that mistake with a birdie at the par-4 11th. Franco notched his final birdie at the last to move into third place after a third-round 68.
Furyk, who had taken the lead with an eagle at the eighth, struggled with his putter on the back nine.
He three-putted for bogey at the 13th and 14th to fall off the pace. He also lipped out a birdie attempt on 15.
Furyk was able to recover one of those strokes with a birdie at the par-4 16th to finish with a 3-under 69 on Saturday.
Billy Mayfair was joined in a tie for sixth place by a pair of Nationwide Tour graduates, Rod Pampling and Cliff Kresge, at 11-under-par 205. Mayfair had a stellar run of five birdies and an eagle in a six-hole stretch starting at the fifth.
Robert Gamez and John Senden are one stroke further back at minus-10.
Matt Peterson, who is looking to become the first Monday qualifier to win a PGA Tour event since 1986, had taken the lead for a while thanks to a great front nine. He birdied the second, then ran off three straight birdies from the fourth to move to 10-under.
Peterson's birdie run was snapped with a bogey at the seventh. He responded with three straight birdies from the eighth to give him a share of the lead. He briefly took the lead outright at 13-under with a birdie at the 12th, however, he fell out of the lead after he double-bogeyed the 13th and 16th. He also bogeyed the 18th to fall into a tie for 14th place at 8-under-par 208.