The result puts Britain 2-1 ahead and means they need just one point from Sunday’s single rubbers to win the tie.
The tone of the match was set in the opening games as Murray raced through the first on serve before the visitors took advantage of early Pavic nerves to break for a 2-0 lead.
Murray and Fleming were rarely tested as they raced to a two-set lead but the Croatian pairing did offer resistance in the third, breaking twice late on and saving match point before clinching a tie-break 8-6.
But it only delayed the end result as Britain quickly established themselves in the fourth set which they took comfortably 6-1.
Afterwards Fleming said: “There were three high quality [sets] even the third was good but we got off to that early break lead in the fourth we seemed to cruise from there.”
Captain Leon Smith added: “The win today was really important. On clay it was definitely the best option. Going into tomorrow it now gives us two cracks at it. I’m very confident Andy can come through and hopefully back into the World Group for the first time since 2008.”
Sunday's singles matches see Murray face Dodig and Dan Evans meet Borna Coric.