Teenager Coric, who recently claimed the US Open boys' title, tested Murray throughout the first set, pushing the Brit all the way before being broken in the eighth game - losing the first set 6-3.
Murray cruised through the second set, punishing the 16-year-old whenever he saw fit.
To the delight of the home fans in Umag, London-based Coric went 3-1 ahead in the third set, breaking the Briton's serve with an audacious flicked winner after chasing down a drop shot, before Murray stepped up a gear clinch victory.
"The first five or six games were tough," said Murray, who is playing Davis Cup for the first time in two years as Britain try to return to the World Group.
"He was solid, he fought for every point. When he gets physically stronger, he's going to be a very tough player. The future is obviously very bright for him."
In the day's second rubber Croatia's number once wasted little time in drawing the two sides level as he put Evans to the sword.
Britain's number two, who secured two stunning victories against Kei Nishikori and then Bernard Tomic at this year's US Open, struggled against a far superior opponent as he fell 6-3 6-2 6-3.
Murray is now expected to partner Colin Fleming in Saturday's doubles, although Jonny Marray could be drafted in to replace either player.
"1-1 is certainly not a surprise after day one - it is where we expected and both favourites won," said British team captain Leon Smith.
"Whoever we put out to play in the doubles I'm confident we will do well and it will be the key rubber."