The 29-year-old was one shot back from overnight leader Sam Burns going into the final day but he held his nerve to finish on 13 under par.
Ortiz held back the tears as he delivered the win with a 20-foot birdie putt, a victory which means he will qualify for the 2021 Masters in April.
“I’ve played great this week and it was really hard to hold the emotions all the way to the end, but I’m really happy the way it played out and the way I played, too,” Ortiz said after winning at the Memorial Park Golf Course.
“I wasn’t really thinking about the other guys, I wasn’t worried. I knew if I played good I was going to be hard to beat.”
Johnson, who is returning to the sport after a positive Covid-19 test, pushed Ortiz to the very end and had missed a putt which would have seen him share the lead with the Mexican.
Meanwhile, Matsuyama mounted a late surge and recorded a course record 63 to put himself into contention but neither man could match Ortiz, who claimed the biggest win of his career.
Fans cheered Ortiz’s triumph, a sight not seen at many sporting events across the world this year.
The Houston Open, which began on Thursday, was the first PGA Tour event on US soil to have fans in attendance since March with 2,000 tickets sold for each round of the tournament.