The Croatian, ranked 13th in the world, went down 3-6 7-6(0) 6-4 to his Romanian opponent, ranked 221st.
Copil crashed down 21 aces and saved seven of 11 break points - Cilic saved only two of six - to earn a match against Florian Mayer after the German beat Fabio Fognini of Italy 6-1 6-2.
Spain's Fernando Verdasco, the eighth seed, also went out as he was beaten 5-7 6-3 6-4 by American Sam Querrey who saved six break points in the final two sets.
Sixth seed Alexandr Dolgopolov of Ukraine beat Guillermo Garcia-Lopez of Spain 6-0 6-4 to set up a second-round clash with either Jurgen Melzer or Pablo Andujar.
Carlos Berlocq will be Novak Djokovic's opponent in round two should the Serb move past qualifier Michael Berrer. The Argentine disposed of Australian Marinko Matosevic 3-6 6-2 6-3.
And Zhang Ze won his all-Chinese clash with Wu Di 6-4 6-1 and Kevin Anderson beat Brian Baker 5-7 6-3 6-1
In Tokyo, former champion Tomas Berdych eased his way into the second round of the Japan Open with a 6-1 7-6 victory over France's Benoit Paire.
Second seed Berdych almost inflicted the dreaded 'bagel' on Paire in the first set until the Frenchman finally held serve in the sixth game but the Czech was pushed in the second.
Berdych, who beat Juan Martin del Potro to win the 2008 Tokyo title, took the second set on a tie-break 8-6 to close out the match in blazing sunshine.
"I was actually surprised at how the first set went," Berdych said. "He made me work harder in the second set.
"I'm taking anything for granted," added Berdych, who defeated Roger Federer in his run to last month's US Open semi-finals.
"I hope if I can continue to play like this it will qualify me for London."
The Czech is currently seventh in race to make the elite eight-man World Tour Finals having qualified for the season-ending shootout the past two years.
"I'm going match-by-match," said world number six Berdych, seeking his second title of the year this week after winning in Montpellier in February.
"I have nice memories here. I like coming back to Tokyo. It was one of the biggest tournaments I won in the past. It gives me positive thinking."
Switzerland's Stanislas Wawrinka was given a fright by Japanese wildcard Hiroki Moriya before the seventh seed came through 7-5 4-6 6-4.
US Open champion Andy Murray opens the defence of his Tokyo title against Frenchman Gael Monfils on Tuesday. Lukas Lacko of Slovakia thrashed Spain's Alberto Ramos 6-0 6-2 to set up a clash with either Murray or Monfils.
Serbia's Viktor Troicki was a man in a hurry, firing 12 aces in a 6-2 6-3 win over Dutchman Robin Haase 6-2 6-3 that took just 64 minutes.
- Tomas Berdych