The Bryan brothers are the most successful pairing in the open era, with the American twins winning a total of 16 Grand Slam titles and more than 100 tournaments in total.
But 14th seeds Inglot and Mergea produced a brilliant performance on Hisense Arena on Monday, rallying from 4-1 down in the opening set to win it in a tie-break and then taking the second 6-3.
"Beating the Bryan brothers on a big stadium like this in front of a huge crowd, it does not really get any better than that," said the 6ft 4in Inglot, who reached the quarter-finals here last year with former partner Treat Huey.
Mergea added: "It was a great experience today. I've played the Bryans a few times and they are the legends when it comes to doubles. It was a good match and we are really happy right now."
Inglot and Mergea will face sixth seeds Jean-Julien Rojer and Horia Tecau in the quarter-finals, but Britain's Jamie Murray and Australian John Peers had earlier lost in three sets to fourth seeds Ivan Dodig and Marcelo Melo.