Djokovic, playing only his fourth tournament since Wimbledon last July after struggling with a right elbow injury, overcame some late nerves to win 7-6 (7/2), 7-5 in the second-round clash.
"It was a real battle. I mean, I could have won the match earlier definitely, but it was great to deal with these kind of emotions again," said the 12-time Grand Slam champion.
Djokovic is working with long-time former coach Marian Vajda for the first time in almost a year after splitting from Andre Agassi, and the 30-year-old has looked rejuvenated in Monaco after early exits in Indian Wells and Miami.
The Serbian will next face Austrian fifth seed Dominic Thiem with a possible quarter-final against defending champion Nadal, who plays Karen Khachanov in the last 16 on Thursday, up for grabs.
But Djokovic insisted he wasn't looking past the clash with Thiem, a French Open semi-finalist in each of the last two years.
"It's disrespectful towards Dominic and Dominic's tennis... To think about and talk about potentially a match-up with Nadal," Djokovic added.
"Last year in Roland Garros he beat me in straight sets. He's definitely (a) top-three player in the world on clay."
Djokovic came through a tight first set in a tie-break and moved a break clear in the second, before late drama as he missed nine chances to close out the match.
The ninth seed rediscovered his composure, though, to finally complete the job at the 10th time of asking as the 39th-ranked Coric sent a backhand flying long.
Two-time Monte Carlo winner Djokovic said that he has gained power since changing to a lighter racquet to try and take pressure off his troublesome elbow.
"It feels like I gained more power, especially on the serve. I gained more angle," he said.
"The racquet is the most important tool that we have, but it's not more important than our body and our mind, our movements. So it is a tool that helps, but it doesn't play for you."
Nadal makes 'positive' start
Nadal started his bid for an 11th Monte Carlo title with a comfortable victory over Slovenia's Bedene.
The 16-time Grand Slam winner, like Djokovic, is looking for a record 31st Masters title, and he swatted aside Bedene for the loss of only four games.
The 31-year-old Spaniard said he still needs more time on court after playing his first tour match since an injury-hit Australian Open quarter-final exit in January.
"It's a positive start, of course. It's important for me to win matches... I need days on court," said Nadal.
"Since Shanghai last year, I (have) spent not enough time on court competing.
"Every match that I have the chance to win is great news because it's confidence and is a chance to play another day, and that's what I need."
Japan's Kei Nishikori continued his comeback from injury with an impressive 7-5, 6-2 win over Russian Daniil Medvedev.
The former world number four, who has struggled over the last year with a right wrist injury, followed up his first-round victory against Tomas Berdych with a solid display on centre court.
"The first set was a real battle, he was playing a little better than me I think at the beginning," said Nishikori.
"I just tried to stay calm and not do anything too crazy and stay calm at the baseline."
Second seed Marin Cilic safely negotiated a potentially-tricky opener against Fernando Verdasco, winning 6-3, 7-6 (7/4).
The former US Open champion will take on Canadian Milos Raonic in round three on Thursday.
Russian Khachanov cruised past Gilles Simon 6-2, 6-2, while last year's runner-up Albert Ramos-Vinolas slumped to a 6-4, 6-2 defeat by Philipp Kohlschreiber.
Italian qualifier Andreas Seppi came from a set down to beat Guillermo Garcia-Lopez and progress to a tie with Nishikori.