Carreno Busta took Rafael Nadal's spot at the ATP Finals after his fellow Spaniard withdrew due to a knee injury, ending the year a career-high 10th in the rankings.
However, the second seed lost out in a first-round scrap against Coric, who twice stopped his opponent serving out the match and saved three match points to claim a 5-7 6-2 7-6 (10-8) triumph.
"It was such a tough match. I think he was serving two times for the match and I was just thinking to myself [that] I was playing good for the whole match and that was my main focus," said Coric.
"I think it's actually quite normal to play better when you’re playing against better players."
Third seed Tomas Berdych, making his return from a back injury that has kept him out since October, was sent packing by Jan-Lennard Struff 6-4 1-6 7-6 (7-4), and Albert Ramos Vinolas and Filip Krajinovic - seeded fourth and sixth respectively - also suffered premature exits.
Ramos Vinolas took the first set against Guido Pella but wasted a chance to serve out the decider before succumbing to a 2-6 7-5 7-5 loss, while Paris Masters finalist Krajinovic was comfortably beaten 6-4 6-1 by Peter Gojowczyk.
Feliciano Lopez saved match point to overcome Guillermo Garcia Lopez 6-4 5-7 7-6 (7-3), while Richard Gasquet dropped just two games in seeing off Victor Estrella Burgos in 52 minutes.
There were also wins for Gael Monfils, Mirza Basic, Aljaz Bedene, Matteo Berrettini and Stefano Travaglia.