Fearnley was ranked outside of the world’s top 500 just 12 months ago but the 23-year-old has risen rapidly up the rankings to a career-high of 55.
He will face either Christopher O’Connell of Australia or French 22nd seed Ugo Humbert in the second round.
After clinching the opening set via a tie-break, Fearnley broke early in the second and raced out to a 4-1 lead before serving the set out to love.
Wawrinka, who knocked Andy Murray out in the first round last year, dropped serve immediately in the third set but fought back to move level at 2-2 – much to the delight of the crowd on court 14.
However, their joy was short-lived as Fearnley quickly restored his lead and won four straight games to wrap up victory.
Fearnley has now won on his main draw debut at each of the three Grand Slams he has featured at so far – Wimbledon, the Australian Open and Roland Garros.