The meeting came as divisions grow in Europe over the proposed tariffs
Olympic bronze medalist Kuchar opened birdie-birdie at TPC Scottsdale, but he did his real damage coming home with three birdies and an eagle at the par-five 13th, where he rolled in a 20-footer.
He surged past Japan's Matsuyama, who was in the clubhouse on six-under 65, where he was later joined by Steele.
"I drove the ball really well, which around here you have to do," Kuchar said. "From there I just played good, steady golf. (I) had a lot of chances.
"It was a fun round of golf," he added. "To play in front of so many people to get the energy and the buzz that's out here, it was fun."
Kuchar is making his first start since the unofficial Hero World Challenge in early December, but showed no signs of rust.
Matsuyama, meanwhile, has powered into 2017 with victories at the Japan Open and World Golf Championships HSBC Champions in October, followed by wins at the Taiheiyo Masters in November and the Hero World Challenge in December, opened the calendar year with a runner-up finish to Justin Thomas at the USPGA Tournament of Champions last month.
He cooled off with 27th and 33rd place finishes at the Sony Open and Farmers Insurance Open. While he's not at the level he was in October, he said, he's hoping he can get there.
Certainly Matsuyama looked sharp in picking up where he left off last year with a birdie at his opening hole, the par-four 10th.
"I wish I knew why I play well here, but I did hit the ball very well today - hit a lot of greens, a lot of good shots," he said.
Steele had twice gained the solo lead at seven-under, but back-to-back bogeys dropped him back to five-under before he gained a share of second with a birdie at the eighth, his penultimate hole. It was his seventh birdie in a round that also included a hole-out for eagle at the 14th.
South Korea's An Byeong-Hun was among a group of five players tied for fourth on five-under 66, along with Americans John Peterson, Robert Garrigus, Scott Brown and Chris Kirk.
Rickie Fowler, beaten by Matsuyama at the fourth playoff hole in Phoenix last year, headed a big group on four-under 67 that also included J.B. Holmes, Steve Stricker, Brooks Koepka, Alex Ceijka, Martin Laird, Webb Simpson and Graham DeLaet. South Korea's Kang Sung was four-under through 17 when darkness halted play with nine players still to finish.
Five-time major winner Phil Mickelson, a fan favorite at the rowdy Phoenix event he has won three times, headlined another big bunch on three-under 68.
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