There were concerns the Wimbledon champion could miss Saturday's rubber because of a back injury but, apart from the occasional grimace, the Scot looked in fine form as he and Fleming recorded a 6-3, 6-2, 7-6 (8/6), 6-1 victory.
Victory for Murray over Dodig in Sunday's first reverse singles rubber would see Leon Smith's team return to the elite World Group for the first time since 2008.
With the teams sharing Friday's singles matches, early nerves in Saturday's rubber resulted in a double fault from Pavic which helped the British pair to break as early as the second game.
That was all the advantage they needed to take the first set, and another break followed in game three of the second set thanks to a winning forehand from Murray.
Britain were firmly in control of the contest at this stage and an error from Dodig in the seventh secured a double-break with Fleming then serving out for a two-sets lead.
The third set seemed to be going the same way as Britain drove home their advantage with another break on Pavic's serve in the opening game.
However, Croatia broke back immediately to roars of approval from the home crowd and, although a winning volley from Fleming secured a further break for the visiting nation, that was wiped out once more to take the third set into a tie-break.
Again, it was Britain who took control of proceedings, winning three points in succession to open up a 4-2 mini-break.
But this time Croatia made the most of the home support and first held off a match point before a double fault from a nervous Fleming allowed Dodig and Pavic to pull a set back.
However, any fears over the British pair - and Fleming's confidence in particular - were quickly brushed aside as some more woeful serving from Pavic helped them to open up a 3-0 lead at the start of the fourth.
This time Croatia had no response and Dodig was broken in the sixth game, allowing Fleming to serve out for the win in two hours and 41 minutes.