Steven Finn and Stuart Broad both took 2-47 as England produced a stifling fielding display to dismiss Australia for 257-9 and take a 1-0 lead in the five-game series.
“I think we were quite comfortable,” Morgan said. “The only worry was the wicket was a lot better with the sun that had been on it in the afternoon, but the control the guys have in the team is fantastic and they bowled really well.”
Australia captain Michael Clarke hit 61 from 67 balls with a six and six fours, while David Warner made 56 from 61 balls, also with one six and six fours, but after crusing to 96-1 the tourists lost wickets at regular intervals and fell too far behind the run rate.
“England deserve a lot of credit, they played really well,” Clarke said. “We probably let them get 20 runs too many and then we lost wickets in our innings at crucial times unfortunately.”
Earlier, England posted 272-5, with Morgan's innings, which included four sixes and five fours, ultimately proving the difference between the sides.
Jonathan Trott hit 54, with three fours from 70 balls, while Clint McKay was the pick of Australia's bowlers with figures of 1-43, after the tourists won the toss and chose to field in overcast and damp conditions.
England captain Alastair Cook admitted he would have bowled given the choice and the openers found life difficult early on.
Ian Bell was on 3 when he was given out caught behind off McKay, but the batsman successfully referred umpire Richard Kettleborough's decision. Replays showed the ball brushed Bell's trousers.
After three separate rain delays, which kept the players off the field for exactly an hour in total, conditions improved, allowing Bell and Cook to put on 74 for the first wicket, before the former was lbw to Brett Lee for 41 at the end of the 17th over.
Bell didn't realise he'd got the faintest inside edge on the ball and walked without asking for a referral.
Cook was dropped by wicketkeeper Matthew Wade from the very next delivery from Shane Watson, but he fell for 40, caught behind off Pat Cummins to the final ball of the 21st over.
Ravi Bopara made just 15 when he edged McKay to Clarke at first slip in the 29th over.
Australia then wasted its referral when Trott was given not out either stumped or caught behind to Xavier Doherty in the 32nd over.
Cummins dropped a difficult return catch off Trott and Morgan hit the first six of the innings with the final ball of the same over, the 38th, by smashing Cummins back over his head.
Trott brought up his 50, from 65 balls, by glancing Lee to backward point for two, but he was bowled by Doherty with the first ball of the 41st.
Morgan reached his 50, from 45 balls, when he nudged Cummins to square leg, then hit Lee for successive sixes from the last two balls of the 47th over as England tried to accelerate.
The next ball that Morgan faced, from Watson, also went for six, but Watson did well to limit England to eight off the final over and had Craig Kieswetter caught for 25 by Warner off the final ball of the innings.
Australia's chase got off to a bad start when Watson was given out caught behind to Steven Finn after a referral but Warner pulled Stuart Broad for six in the 15th over and he reached his 50, from 57 balls, with a single in Broad's next over.
Australia was coasting until James Anderson bowled George Bailey (29), who played on to the final ball of the 19th over.
Anderson had Warner caught by a diving Kieswetter in his next over and with the scoring rate dropping Clarke called for the batting powerplay after 29 overs.
The move backfired when David Hussey played the first ball, from Finn, onto his stumps and was bowled for 13.
Clarke was on 13 when Tim Bresnan dropped a return catch, but Bresnan then had a fidgety Steve Smith caught behind for 8.
By the end of the powerplay Australia had added just 17 runs for the loss of two wickets.
Australia went 10 overs without a boundary until Clarke edged Broad through the vacant slip area for four.
Matthew Wade pulled Anderson for six at the start of the 41st over and Clarke brought up his 50 by smacking Swann for six in the next over, but two balls later when Clarke tore down the wicket for an unlikely single, Wade was run out by the substitute fielder Jason Roy.
In the 43rd, Bresnan traped Clarke lbw with a full toss and McKay perished in the 46th when he holed out to Broad and Kieswetter took a steepling catch.
Cummins unsuccessfully referred a Broad lbw and was out for 4, and despite a battling 29 not out from Lee, England clinched the win when Australia failed to hit the 21 runs needed from the final over.
The second match is on Sunday.