Johnson, who won his first major at the 2007 Masters, seeks his 13th career PGA title and third Texas Open victory after 2008 and 2009 at LaCantera Golf Club, where he shared the course record of 60.
Pars from both men on the par-five 18th meant it was the Texan native who lifted his maiden PGA Tour title, while Mullinax would finish in a share of second after Sean O'Hair chipped in for a birdie on the final hole.
The 30-year-old began the day in a tie at the top with Zach Johnson, but quickly assumed the outright lead with birdies on each of his three opening holes. He added birdies on the sixth and 10th holes before bogeying the 11th. Landry, interviewed on camera after he had finished 21st on the 2015 Web.com Tour money list, was asked which players he looked up to.
Johnson was vying to become the third player to win the Valero three times, after Arnold Palmer (1960-'62) and Justin Leonard (2000-'01, '07).
"It was obviously a tough week this week", Landry said. "And you've still got to hit good shots and putt".
Mullinax kept up the pressure throughout the round, making a nifty par from above the bunker in the middle of the green on the par-3 16th to stay within one shot.
Jimmy Walker, the 2016 Valero Texas Open victor and a local favorite, shot 5 under on Sunday and finished alone in fourth at 14 under.
Landry took a two-stroke lead to the par-5 18th after Mullinax chunked a flop shot and bogeyed the short par-4 17th. Second-round co-leader Ryan Moore (even par of Sunday) was another shot back in seventh.
"It's incredible. It's a lot of fun to be standing among the best players in the world on that leaderboard", Niemann said. "Before this week, my game was good, but my putter was not good enough, but that changed here". "Just try to keep it up and hope to play well for the next weeks".
Meanwhile, former amateur world No 1 Joaquin Niemann enjoyed a superb first tournament as a professional as he birdied the final three holes to card a 67 and post a sixth-place finish on 12 under par. He played well in the second and third rounds as well, pairing with Shane Lowry of Ireland in the final group in the final round and eventually finishing tied for 15th.
"I didn't hit that great a wedge shot", Landry said of his approach. "We still have a lot of work to do, but this will definitely help".
The nearest Landry had been to victory was in January when he lost to Spain's Jon Rahm on the fourth playoff hole at the PGA Careerbuilder Challenge. He is slated to tee it up again next week with Talor Gooch in the Zurich Classic team event.