The 39-year-old wild card Wawrinka, formerly ranked number three in the world but now down to number 79, was no match in the first round for the Australian who prevailed 6-3, 6-0 on the main Rainier III court.
Wawrinka, the 2014 champion in Monte Carlo and a two-time Grand Slam winner, made 28 unforced errors to De Minaur's 18 and landed just four winners to the 12 made by his opponent.
In the next round, De Minaur, who won this year's title on the hard courts in Acapulco, will face Dutchman Tallon Griekspoor.
Persistent rain meant that the scheduled start at 11:00 local time (10:00 BST) was delayed by nearly two and a half hours.
More rain was expected in the Principality which would play havoc with the schedule.
If the weather allows, world No.1 Novak Djokovic is due on court later on Tuesday to face 41st-ranked Russian Roman Safiullin.
