Wawrinka looked in danger of falling in his opener for a second straight year after losing a tight opening set, but found a way back into the match by dominating the second and grabbed an early break in the decider.
Griekspoor beat Wawrinka in their only previous meeting in 2018 and showed plenty of fight to level at 3-3 but surrendered his serve again to allow the Swiss veteran to surge ahead and close out the contest on his sixth matchpoint.
"It was really important to stay calm with myself," said 2014 champion Wawrinka, the oldest man in this year's field.
"In the first round you need to find your game. The first match on clay is never easy against a good player. I'm happy to get through. It was important to fight until the end."
Up next for the three-times Grand Slam champion is world number 10 Taylor Fritz (25).
"He's an amazing player," Wawrinka said of the American. "He's top 10, been winning a lot of matches. Winning a lot of tournaments. He's very dangerous on many surfaces."
Murray marches...out
Murray gained direct entry into the prestigious tournament after several players including world number two Carlos Alcaraz (19) and 11-times champion Rafa Nadal (36) withdrew due to injuries.
But the former world number one endured a torrid time in a rare appearance in a claycourt tournament due to hip injuries as he lost 6-1 6-3 to Australian De Minaur in 86 minutes.
World number one Novak Djokovic (35) will return to the tour on Tuesday having been forced out of tournaments in Indian Wells and Miami last month after being denied special permission to enter the United States without a COVID-19 vaccination.
The 22-times Grand Slam champion last played in early March in Dubai and has been ramping up his preparations on clay ahead of the French Open which begins next month.
He will meet Russian qualifier Ivan Gakhov (26) who beat American Mackenzie McDonald (27) 7-6(6) 2-6 6-3.
Francisco Cerundolo (24) earlier stunned 11th seed Cameron Norrie (27) 6-3 6-4 and will play Matteo Berrettini (26) who downed Maxime Cressy (25) 6-4 6-2 for his first win in the tournament.
Thiem downs Gasquet
Former US Open champion Dominic Thiem beat French veteran Richard Gasquet 6-1 6-4 as the Austrian looks to revive his career following his slide out of the top 300 due to a wrist injury.
Thiem, now ranked 106 in the world, had split with coach Nicolas Massu before the tournament, ending a partnership that lasted more than four years.
But the Austrian looked like he was slowly getting back to his vintage best as the claycourt specialist won five games in a row at one stage while he did not face a single break point in the match.
The victory, sealed with his sixth ace in the contest, also marked Thiem's first at a Masters tournament since Rome in 2021, weeks before the wrist injury that sidelined him for nine months.
Thiem will next play Danish Holger Rune, who is ranked ninth in the world.
