Sanchez equalised after Craig Dawson's 12th-minute opener but substitute Hal Robson-Kanu made an instant impact 10 minutes after half-time and Dawson headed a second to ramp up further pressure on the beleaguered Wenger.
Fan protests, both with banners in the away end at The Hawthorns and on the back of a light aircraft circling the ground - a rival plane had a message of support for Wenger - were again a feature as the 67-year-old lost his top scorer during Arsenal's fourth defeat in five Premier League games.
Sanchez was on the end of a heavy challenge from West Brom winger James McClean at the end of the first half and was visibly hampered until he made way for Alex Iwobi in the 78th minute, compounding a miserable outing that also featured goalkeeper Petr Cech limping off with a calf injury before the interval.
"We lost Petr Cech in the first half with a calf problem and after we lost, of course, Sanchez," he told Sky Sports.
"He stayed on but in the second half he was not himself so I decided to take him off. His ankle doesn't look good."
Sanchez must now be considered a doubt for Chile's forthcoming World Cup qualifiers against Argentina and Venezuela, although Wenger told a post-match news conference the former Barcelona man insisted on soldiering on after half-time at West Brom.
"It's a bad tackle and you see his ankle is in an absolutely terrible state," he said.
"He should not have played in the second half but he insisted that he wanted to come out.
"In the first-half he was the one who created many chances and many dangerous situations. In the second half you could see he couldn't anymore. In the end he limped [off].
"We had to take him off. He shouldn't have even played in the second half."
Sanchez's participation for his country is likely to hinge upon whether he has suffered ligament damage.
"Chile go to Argentina for a massive game but they play only next week," Wenger added.
"If it is only a kick he will play. His ankle is swollen. If a ligament is damaged he will not play."