Johnson and Palmer, the top seeds, clinched a place in the semifinals with the victory, having opened play in the Touchtel Group on Tuesday with a three-set win over Belgium's Tom Vanhoudt and South Africa's Chris Haggard.
Johnson and Palmer captured Wimbledon and six other titles in 2001 and already earned crowns at Doha and Sydney in 2002.
The 23-year-old Bryan twins, who wasted a one-break advantage in the second set, already have revenge on their minds.
"Our level probably slipped a little bit. We could have done a few things," Bob Bryan said. "If we win tomorrow, hopefully we'll play (Johnson-Palmer) in the finals."
The Bryan twins, who won four events in 2001, look to clinch a place in the semifinals on Thursday, when they face Vanhoudt and Haggard.
Vanhoudt and Haggard slipped to 0-2 on Wednesday. American Rich Leach and South African Ellis Ferreira rallied to beat Vanhoudt and Haggard, 6-7 (3-7), 6-1, 6-3.
Leach and Ferreira won the 2000 Australian Open, as well as three Tennis Masters Series events.
Leach and Ferreira meet Johnson and Palmer on Thursday.
India's tandem of Leander Paes and Mahesh Bhupathi also moved into the final four with a 7-6 (7-4), 5-7, 6-3 triumph over Mark Knowles of the Bahamas and Brian McPhie of the United States.
Also in the Bharti Group, the Czech duo of Petr Pala and Pavel Vizner cruised past South Africans John-Laffnie de Jager and Robbie Koenig, 6-3, 6-2.
On Thursday, Paes and Bhupathi meet Pala and Vizner, while Knowles and McPhie go up against de Jager and Koenig.
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