Norrie, the 2021 champion, broke Wu four times en route to the win and will next face either Italy's Matteo Berrettini (26) or Japanese qualifier Taro Daniel (30), who play later on Friday.
"Lots of good memories at Indian Wells," Norrie said. "It's nice to be through to the third round."
The 10th seed is off to a terrific start this year having beat world number two Carlos Alcaraz (19) in the Rio Open final last month to claim his fifth singles title.
Norrie's win comes a day after unseeded Britons Andy Murray (35) and Emma Raducanu (20) reached the second round at the ATP and WTA 1000 event in the Southern California desert with gutsy wins.
Norwegian Casper Ruud's (24) serve-forehand one-two punch proved too much for Diego Schwartzman (30) as the third seed advanced 6-2 6-3 over a player who has given him trouble in the past.
"Everything sort of went my way," said Ruud, who now has four wins and five losses against the Argentine.
"Early on there were a couple of line shots and net cords that went my way, so I kind of felt that maybe today is my day."
On the women's side, Teichmann beat birthday girl Bencic to reach the third round of the tournament for the first time.
Ukrainian Lesia Tsurenko (33) beat Croatian Donna Vekic (26), Latvian Jelena Ostapenko (25) defeated Aliaksandra Sasnovich (28) and Czech Barbora Krejcikova (27) breezed past Ukrainian Dayana Yastremska (22).
