The J-League champions, who have yet to win the continental title in three previous involvements, impressed against their hosts with Erik lashing in the opener from the edge of the penalty area in the 19th minute.
The Brazilian winger turned provider when he delivered an inch-perfect cross from the right for Miyashiro to score with a downward header past Port goalkeeper Chen Wei.
Kobe's third arrived just before half-time and Erik was again involved, releasing Gotoku Sakai behind the Shanghai backline, where the defender pulled the ball back for the unmarked Osako to slot beyond Chen.
Kobe joined fellow J-League side Sanfrecce Hiroshima, South Korean clubs Gangwon FC and Ulsan HD, and Buriram United from Thailand as winners from the opening round of matches in East Asia.
Former champions Ulsan needed a late fightback to secure a 2-1 victory at home over tournament newcomers Chengdu Rongcheng on Wednesday.
The Chinese Super League table-toppers took a 44th-minute lead when Pedro Delgado - also known as De Erjiaduo since gaining Chinese citizenship in 2019 - fired into the bottom corner of Jo Hyeon-woo's goal.
Substitute Um Won-sang levelled for Ulsan with 14 minutes remaining, racing onto Darijan Bojanic's long ball from inside his own half before beating goalkeeper Jian Tao.
Delgado was dismissed when he picked up a second yellow card in the third minute of stoppage time for a foul on Choi Seok-hyun and, two minutes later, Ulsan capitalised when Heo Yool scored the winner.
The first eight finishers in the 12-team league phase of the competition will advance to the knockout rounds in March with the quarter-finals, semi-finals and final to be played in Saudi Arabia in April.