Morgan started his innings cautiously but launched an all-out attack on the Australian bowling in the last 10 overs, smashing four boundaries and clearing the rope six times in a brutal display, his second 50 taking just 24 balls.
It could have been very different for the home side, who thought they had Morgan caught in the deep for just one when he skied a catch to deep mid-wicket off the bowling of Australian captain Michael Clarke.
England captain Alastair Cook won the toss on a hot and humid afternoon and had no hesitation batting first on a benign Gabba stadium wicket.
The tourists got away to a bright start with Cook and Ian Bell pressuring the Australian fast bowlers, mixing some big-hitting with quick singles.
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They put on 57 off only 68 deliveries before Cook drove a ball back to spinner Glenn Maxwell, who dived forward and took a good return catch low to the ground.
Bell and Morgan set about rebuilding the innings, adding 61 runs for the fourth wicket.
Bell was run out with the score on 139 but Morgan then shared good partnerships with Ravi Bopara (24) and Jos Buttler (49) to give themselves a chance of ending a run of six consecutive losses to the Australians.