Murray brushed aside Borna Coric, the 16-year-old US Open junior champion, 6-3 6-0 6-3 but British number two Dan Evans, ranked 149, lost 6-3 6-2 6-3 to world number 35 Ivan Dodig.
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."
However, Murray admitted to suffering from a back problem in his first match since losing in the quarter-finals of his US Open defence in Flushing Meadows.
But he had enough experience to brush aside his teenage challenger on the clay court surface.
"My back is a little bit sore," he said.
"The change of surface is difficult but after a couple more matches I'll have some time to rest, so let's just get the win this weekend and earn our promotion.
"I thought the first five or six games were tough but once I got the break, I won a few games in a row.
"I lost my concentration a bit at start of the third but saw it through. It was alright - it could be better."
Victory over Croatia would secure Great Britain's return to the elite world group for the first time in five years.
� Sportsbeat 2013