The Britons beat Filip Krajinovic and Nenad Zimonjic 6-1 6-7 (2-7) 6-3 6-4.
Earlier, James Ward lost 6-1 6-3 6-2 against Dusan Lajovic in their singles match, which was delayed from Friday.
Kyle Edmund, who won his singles match on Friday, can seal victory for Great Britain when he faces Lajovic on Sunday at 14:00 BST.
Britain are aiming to become the sixth team to defend the Davis Cup title, after they won it in Belgium last year.
Murray was partnering Inglot after his brother Andy, with whom he has won all five of his previous Davis Cup doubles matches, decided to sit out the tie. The British pair started strongly and raced into an early lead against their Serb opponents, who were playing together for the first time.
Despite the Britons having chances to break in the second set, the Serbs improved to level, but Inglot and Murray recovered to put their team within one rubber of the semi-finals.
Inglot, who sealed his first Davis Cup win with the victory after serving out the third and fourth sets, said: “It is great to feel like I have finally put a rubber in for the team.
“Serving out was a little nerve-wracking but with Jamie at the net you know he will help you out. He deserves a lot of credit.”
Murray added: “It was unfortunate we lost the second set but we kept getting the ball back on return and then the breaks of serve helped us.
“We’re really happy to have that win going into Sunday.”
Team captain Leon Smith picked out the Inglot serve as a key in the match.
“It’s great for Dom to have his first Davis Cup victory,” Smith added. “Jamie is really experienced now and it’s great that we can rely on him to come out and do what he does.”
Ward was a late inclusion in the British squad after replacing Andy Murray but he lost in one hour and 52 minutes against a player ranked 159 places above him, at 81st in the world.
The 29-year-old, whose ranking has dropped after he was inside the world’s top 100 this time last year, was broken in the first game of the match and struggled throughout against the highest-ranked player in Serbia’s squad.
In a match postponed on Friday – and again delayed on Saturday because of rain – Ward made 73 errors and hit just 12 winners compared to Lajovic’s 23.
Ward said: “It’s one of the heaviest clay courts I’ve ever played on.
“It definitely doesn’t suit my game. But he was very good today. The tactics we had set up were pretty much completely opposite to everything he did in the first set, which is credit to him.”