American Perez, who moved to a cross-handed putting method two months ago, finished in style with birdies at the final three holes for a four-under-par 68 at the TPC San Antonio.
New Zealander Lee also plundered the closing stretch with four birdies in the final five holes to join Perez, one stroke ahead of American Will MacKenzie and South Korean Noh Seung-yul with half the field back in the clubhouse.
Perez, 38, whose lone PGA Tour win came in 2009, went cross-handed at the San Diego tour stop in January and promptly finished tied second that week.
"I want to putt conventional, but I don't putt well doing that," he told Golf Channel. "So I went back to cross-handed and started to putt well. I got some old feelings back that I had when I won, and I've just taken it from there."
South Korean-born Lee also changed his putting grip recently, going to the "claw" grip, so-called because the player rests his bottom hand lightly on the club and takes most of the wrist action out of the stroke.
"I was hitting it really good but I wasn't making that many putts early in the round but at the end I was making quite a lot of putts," said Lee.
It was a different story for three-times Masters champion Mickelson, who hit only nine greens in regulation en route to a five-over-par 77 that left him in a share of last place with half the field still on the course.
He also struggled on the greens, running up 32 putts and missing several short ones, including a tiddler from barely two feet for a double-bogey at his final hole.
While some players, notably Tiger Woods and Adam Scott, generally prefer to take time off before the majors, Mickelson likes to play his way in.
Therefore, he is making his first appearance at the Texas Open in more than two decades, and is also planning to contest next week's Houston Open.
Morning fog delayed the start of play by nearly three hours.
(Reporting By Andrew Both in Cary, North Carolina; Editing by Frank Pingue)