At six-under 136, they lead by one stroke from Scotland's Russell Knox, who fired a best-of-the-week 65, joint first-round leader Luke Donald, of England, and American Patton Kizzire.
Day, who won back-to-back PGA Tour titles at the Arnold Palmer Invitational and the WGC-Dell Match Play before tying for 10th in mainly brutal conditions at last week's Masters, was delighted to card a two-under round, even more so given his weary state after the year's opening major.
"I felt like I went 10 rounds in a championship fight (last week). It just beat us up, physically and mentally," Day told reporters.
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"Coming to this week, you've got to get yourself back up come Thursday. I may be a little mentally fatigued but there's no excuse. I've got to focus."
And Day had to focus especially hard after hitting his tee shot out-of-bounds on his fifth hole of the day en route to a bogey six.
"Moments like that, it gets to a breaking point. I was playing great and (then) hit one out-of-bounds," he said. "A lot of people don't realise how much the mental side comes into play at this level.
"Right now I'm sitting in a pretty good position for the weekend. I just got to keep doing what I'm doing."
As well as fatigue, he had to cope with fluctuating winds that made club selection difficult at times.
"It was gusting a good 20 to 30 mph (32 to 48 kph) at times and then all of a sudden it would just drop to nothing," he said.
"So it was all about trying to pick the right gust to hit on, being patient enough to stick it out some times."
Former world number one Donald, who shared the first-round lead with South African Branden Grace on 65, double-bogeyed his fifth hole of the day before finding better form after the turn.
"I felt like I didn't quite have it today, especially on the front nine," said Donald, who started his round at the par-four 10th. "I was searching a little bit in my swing.
"But I really grinded it out pretty well, got some good-and-downs and played a really solid front nine coming in."
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Andrew Both)
