Defeat in just over two hours left 2014 US Open winner Cilic with just one win from six matches since January's Australian Open, where he went out in the fourth round.
Pella, who won his first career match at the event in the first round on Sunday, emerged a winner as he played on the idyllic centre court for the first time.
"This is my best surface, I knew I had chances," the world number 35 said. "But I had to play well.
"I was a bit nervous at the start, it was my first time on this court. But playing here is a dream - it's unreal."
Cilic was playing here for the 11th straight year, with quarter-final showings in three of the previous four editions.
Left-hander Pella came to the European clay with a title this season in Sao Paulo. He now stands 13-3 on the surface in 2019.
Top seed Novak Djokovic was set to begin his pre-Roland Garros campaign later, facing Philipp Kohlschreiber, who beat him last month in the Indian Wells third round.
Pella moved into a winning position as he broke Cilic for the eighth time of the afternoon, winning a six-minute game before serving out the upset, saving a break point in the final game and wrapping up as Cilic fired wide.
In the first round, Briton Cam Norrie advanced in his debut match at the event, with the 23-year-old defeating Adrian Mannarino 6-4, 6-3.
The 56th-ranked Norrie was impressed with the dramatic setting of the iconic Monte Carlo Country Club venue which overlooks the Mediterranean from a cliff.
"I like the conditions here. Everything is kind of on top of each other," he said.
"It's pretty sick kind of playing tennis on the side of a mountain with a water view."
Canadian revelation Felix Auger-Aliassime advanced into a second-round contest with third seed Alexander Zverev after beating Argentine qualifier Juan Ignacio Londero 7-5, 7-6 (7/5).
The 18-year-old Auger-Aliassime, who reached last month's Miami semi-finals from a qualifying start, now boasts a ranking of 33rd - up from 175th when he played here a year ago.
Frenchman Pierre-Hugues Herbert defeated Spanish veteran Fernando Verdasco 6-4, 6-4, while local Gilles Simon rounded out opening-round play with a 7-5, 6-1 defeat of Australian Alexei Popyrin.
Popyrin, 19, and playing his first ATP-level match on clay, received medical treatment on his neck while trailing a set and 4-1.