The pair defeated Daniel Nestor and Vasek Pospisil 7-6 (7/1) 6-7 (3/7) 7-6 (7/3) 6-3 to put the visitors 2-1 up in Ottawa ahead of today’s reverse singles.
The tie was delicately poised after the first day following victories for Dan Evans and Pospisil, making what is so often a decisive match arguably even more important.
It was a very even battle on paper, too, with all four experienced doubles players.
Very experienced in the case of 44-year-old Nestor, who received a huge ovation from the enthusiastic crowd at the TD Place Arena as he lined up at his 50th tie.
Pospisil was still in nappies when Nestor made his Davis Cup debut in 1992 but the Canadian remains at the top of the doubles game.
It was no surprise that opportunities for breaks were few and far between, with the unusually fast court favouring the server.
Canada had a chance to take the first set at 6-5, and Inglot’s heart would have been in his mouth when his second serve clipped the tape.
But it dropped in, Britain won the point and then reeled off seven points in a row to take the tie-break.
A brilliant lob from Nestor helped Canada hit straight back with the first break of serve at the start of the second set, an advantage they held until the eighth game.
Overnight repairs to the court meant HawkEye was not available, and Murray was furious that a Nestor serve was not called out on break point.
But Inglot struck a return winner on the next point to ensure it did not matter.
This was a fourth appearance in the competition for the Londoner, who has spent much of his Davis Cup career sitting on the sidelines cheering on the Murray brothers.
Inglot has often looked very nervous, and he and Murray were not altogether convincing in their only previous victory together in Serbia last summer.
Some early mistakes from Inglot, along with a Murray double fault, set the tone in the second-set tie-break but Inglot was arguably the best player across the match as a whole.
He created a chance for a break of the Pospisil serve at 2-2 with a brilliant return only to miss the follow-up volley, and Britain spurned more chances four games later.
But, as in the first set, they made a great start to the tie-break and did not look back.
Pospisil had tape on his left knee after receiving treatment to it on Friday and it was he who was in trouble on serve again in the sixth game of the fourth set.
He saved two break points but dumped a volley into the net on the third and it was fitting that it was Inglot who served out the victory.
Victory for Evans against Pospisil in the first match on Sunday would clinch the tie for Britain, while a win for the Canadian would set up a decisive fifth rubber between Kyle Edmund and 17-year-old Denis Shapovalov.