Berdych found his rhythm early in the match and broke Del Potro for a 3-1 lead and did not lose a point on his powerful serve until the 10th game.
But as Del Potro, a semi-finalist at the tournament in 2012 and 2013, worked his way back into the match, Berdych slowly began to lose confidence and failed to win a single game in the deciding set.
"During the second set, in the end, I played my best tennis," 2009 U.S. Open champion Del Potro told reporters. "I hit so sure with my forehand. I made good serves and I took all my chances to win.
"Mentally, I was strong enough to keep fighting and I think he got down after losing that second set. Then, when I broke his serve early in the third (set), my game started improving and I played much better than him."
Del Potro advances to meet American qualifier Mitchell Krueger in the second round.
Australian Nick Kyrgios also opened his campaign with a welcome victory as the world number 23 beat Belgian ninth seed David Goffin 6-2 6-3 in the afternoon heat to reach the second round.
Other than in the first game when he had to climb out of a 0-40 hole, Kyrgios experienced few problems on serve and even secured three breaks against Goffin, who had to take a medical time out for a knee injury during the match.
"He obviously wasn't at 100 percent today but you know I'm not at 100 percent either, so it was a bit of a battle," said Kyrgios, who has struggled with an array of injuries throughout the season.
"But I served well at crucial moments. I'm feeling okay. I think as long as I can keep putting in performances like these and being able to compete I think I'm okay."
In other action, third seed Dominic Thiem of Austria was barely tested in his 6-3 6-2 second-round win over Italy's Fabio Fognini, while Jack Sock beat fellow American Tommy Paul 6-3 6-3 to reach the round of 16.
Earlier in the day, big-serving Canadian Milos Raonic, the sixth seed, announced his withdrawal from the tournament due to a lingering left wrist injury.
(Reporting by Larry Fine in Tampa, Florida; Editing by Toby Davis/John O'Brien)
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