To remove Morgan, Maxwell took a brilliant one-handed reflex catch at gully with the ball seemingly on an unstoppable path to the boundary.
He went one better to dismiss Liam Plunkett, catching a ball which was destined for six, before flinging it in the air while falling over the rope and recovering in time to claim a fair catch.
Australia were in major trouble at 3-30 when Maxwell arrived at the crease, and he set about repairing the damage with a 137-run partnership with George Bailey (75).
Maxwell produced his full array of conventional and unconventional shots, hitting a pair of booming sixes off Adil Rashid, while his first ball faced from offspinner Moeen Ali was reverse-swept for four.
He ultimately fell to the same shot, bowled around his legs by Ali while attempting one reverse too many.
However, he'd set the platform for Victorian teammates Matthew Wade (50 off 26) and John Hastings (34 off 26) to help push England's target to 300.
Pat Cummins struck early and often for Australia throughout England's innings, finishing with career-best figures of 4-49 including the key scalps of Alex Hales, Jason Roy and Morgan.
Australia had their chances to ice the game - Joe Burns took a sharp catch off Moeen (19 not out), only to fall over the boundary rope and award a six.
Wade also dropped a regulation catch when he grassed an edge from Ali off Cummins. Despite the star performances by Cummins and Maxwell, there was not enough support to lift Australia to victory - with quicks James Pattinson (0-73) and Hastings (0-56) both proving expensive.
After a hot start from Roy (36 off 33) and James Taylor (41), Morgan laid the bedrock of the English innings with his classy innings.
While Maxwell's magnificent catch stopped him short of a well-deserved century, Morgan had put in place the foundations for Ben Stokes (41), Jonny Bairstow (31) and Moeen to guide them to victory.