The world's best pairing had seen a two-set lead disappear as their opponents produced some inspired tennis, but experience eventually told in the final set for USA to prevail 6-3 6-2 3-6 6-7 9-7.
After James Ward's heroics against John Isner on Friday night, the Emirates Arena was rocking as they relished the prospect of Great Britain winning the tie with a day to spare, but the Bryan siblings quickly showed their intent to throw a spanner in the works.
The legendary doubles duo broke the serve of Inglot and Murray twice in both the opening two sets to secure a commanding lead as well as deflate the home support.
However, perhaps inspired by the fighting spirit of Ward, the British pairing hit back to take the third with a single break of serve to suggest that another fairytale comeback could be in the offing.
Midway through the fourth, USA wasted three break-point opportunities, and they were left to rue their missed chances as Inglot and Murray held their nerve to win a marathon tie-break.
The decider saw both teams perform nervelessly on serve, with no break points being created in the opening 14 games, but the Bryans broke the resolve of Inglot and Murray in game 15 to take the tie into Sunday.