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World number 14 Goffin had lost his previous three matches heading to Monaco, but the Belgian delivered a sharp performance on Sunday, winning 6-4 3-6 6-0.
Open Sud de France champion Goffin did not start well, and had to come from 3-0 down in the first set, but Cilic made 52 unforced errors to help his opponent progress.
"It’s a good win. [This is my] first tournament on clay [in 2021]. Starting with a victory is good," said 2017 Monte Carlo semi-finalist Goffin, quoted on the ATP website.
"I did what I needed to do to win that match. I was playing better and better. In the first and second set, I had problems with the returns. In the third set, I was more consistent. It was a good match."
Goffin will face Marco Cecchinato or Dominik Koepfer in the second round, where he has been joined by Australian Jordan Thompson.
Thompson, the world number 62, needed three hours and two tie-breaks to beat Paire 6-4 6-7 (3-7) 7-6 (7-5), as the Frenchman's dismal string of results in 2021 continued.
Paire has now won just one of 10 matches this calendar year, while Thompson registered a fifth win of the season.