The second-seeded Tsitsipas had his shoulder taped and the injury appeared to affect his backhand.
Jordan took advantage and attacked the Greek's backhand while charging in on short balls to earn his second career win over a top-10 player.
He polished off Tsitsipas, the Australian Open runner-up, in two hours, 37 minutes.
"I felt confident, even playing against a top-10 player I felt comfortable in my own skin and what I was doing, and just snuck out a win," Jordan said. "It was a great feeling. You could see it on my face at the end of the match. I was overcome with joy."
Rain delayed play at Indian Wells Tennis Garden and a few fans jumped into action to help dry the hardcourts.
No. 3 seed Casper Ruud cruised past Diego Schwartzman 6-2, 6-3. Ruud raced to a 4-1 lead in the first set and won the only break point on his serve with a forehand that clipped the line.
No. 10 seed Cameron Norrie advanced to the third round with a 6-2, 6-4 win over qualifier Wu Tung-Lin of Taiwan.
A handful of seeded men fell. Jason Kubler moved on when No. 21 Grigor Dimitrov retired trailing 2-6, 7-6 (5), 3-0. No. 22 Roberto Bautista Agut lost to Emil Ruusuvuori, 7-6 (5), 6-2. Ilya Ivashka advanced when No. 28 Botic Van de Zandschulp had to retire trailing 7-5, 3-2. No. 29 Yoshihito Nishioka fell to qualifier Cristian Garin, 6-4, 6-0.
In women's play, No. 2 Aryna Sabalenka defeated Evgeniya Rodina, 6-2, 6-0.
No. 6 Coco Gauff routed Cristina Bucsa, 6-2, 6-4.
No. 9 Belinda Bencic lost to Jil Teichmann, 3-6, 6-3, 6-3, and No. 29 Donna Vekic was beaten by Lesia Tsurenko, 2-6, 6-2, 6-2.
No. 11 Veronika Kudermetova defeated Anna Blinkova, 6-3, 6-4. Sixteenth-seeded Barbora Krejcikova beat Dayana Yastremska, 6-1, 6-2. No. 24 Jelena Ostapenko got by Aliaksandra Sasnovich, 7-5, 3-6, 6-2.
Rebecca Peterson moved on when No. 22 Zhang Shuai retired at 3-0 in the first set. Indian Wells, March 11 (AP/PTI)