Moeen Ali (72 not out) and fellow left-hander Morgan put on 135 in just 75 balls for the third wicket.
They came together after England, sent into bat, had lost openers Alex Hales (three) and Jason Roy (11) in four balls from paceman Pat Cummins to be 18 for two in four overs.
But Australia then fought back well to keep England to under 200.
Hales played over the top of a fast and low full toss before Roy's miscued drive was well caught by Nathan Coulter- Nile, running back from mid-off.
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But Ali, with a cover-driven four off Mitchell Marsh plus a pulled six off Australia debutant Marcus Stoinis, got England going again.
Morgan then weighed in with a superb straight six off all-rounder Marsh that sent the ball soaring out of the ground and into the River Taff.
England then took 19 runs off leg-spinner Cameron Boyce's lone over of the innings. Ali struck him straight down the ground for six and next ball pulled him through midwicket for four to complete his fifty.
Morgan's straight six off Watson brought up a century stand in 61 balls, with the veteran all-rounder's two wicketless overs costing 33 runs.
The captain pulled Watson, one of eight bowers used by Australia captain Steven Smith, for six as he went to fifty.
He faced 39 balls, including three fours and seven sixes, with England now 153 for three in the 17th over.
Cummins, however, came back well to finish with two for 25 in his maximum four overs.
Ali, deprived of the strike late on, faced 46 balls including six fours and two sixes.