Michael Clarke’s men posted a target of 227 all out and appeared to be edging towards a win when Clint McKay’s hat-trick dismissed Kevin Pietersen, Joe Root and Jonathan Trott early into England’s innings.
However, Buttler hit 65 not out off 48 balls, to help the hosts secure victory, while captain Morgan and Michael Carberry produced a valuable stand of 108 runs to play their part.
“There was some hard-fought cricket, especially with the bat,” Morgan told BBC Sport.
“Our backs were against the wall. Me and Carbs dug in, we didn’t expect it to nibble around all through the innings.
“There’s a lot of positives from today. Michael Carberry and Jos Buttler was world class at the end.
“Carberry was cool and calm. He’s got a great head on his shoulders.
“It was good for me to get through a tough spell and set it up at the back end.
“Ben Stokes has taken on a huge amount of responsibility and come up trumps. He’s been one of the finds of the series.”