The Swiss faced match point at 5-6 in the fourth-set tiebreak but won it 10-8 and then raced through the final set to clinch a 4-6 6-3 6-7(6) 7-6(8) 6-2 victory.
"It was an unbelievable fight," Wawrinka said. "I'm for sure lucky to get through that match, saving match point. He played really great, he's really talented and he was really pushing me.
"To finish like that, I'm really happy to get through. It wasn't easy for me to find my best game."
World number 64 Evans, who was ranked as low as 772 in May of last year, stunned Wawrinka as he took the match to the two-time grand slam champion in a packed Louis Armstrong Stadium.
The 26-year-old, trying to reach the last 16 of a grand slam for the first time, broke in the 10th game to take the first set and though Wawrinka hit back to level the match, the Briton took the third in a tiebreak.
The pair traded breaks at the start of the fourth set but Evans, who surprised Wawrinka with his variety, had match point in the tiebreak as the crowd rose to their feet in anticipation of a massive upset.
But Wawrinka saved it with a volley winner, and that proved the last gasp for Evans as the Swiss, a semi-finalist in New York in two of the past three years, prevailed in the tiebreak and then dominated the fifth set.
The Swiss will next play the winner of the match between Australian Nick Kyrgios and Ilya Marchenko of Ukraine.
(Editing by Andrew Both)
