Chris and I tried to push on, we were looking at 270 but we had a couple of good overs.
Steve Smith will stay at No.4 for the remainder of the one-day series against England, but the Australian captain says he needs to recapture his own form at the crease.
Buttler and Woakes together hit 102 runs from the final ten overs of the game to get England up and past 300.
Several English batsmen failed to capitalise on promising starts, but Buttler notched his fifth one-day worldwide century from the last ball of the innings as England helped themselves to 38 off the final two overs. Australia were left wondering what hit them.
Jos Buttler took the Aussie attack apart.More news: Greeks rally against use of 'Macedonia' in name dispute with Skopje
With the series now wrapped up 3-0, it seems unlikely the 32-year-old will return for the visitors in the next week. In their last five overs England had made 66 after being 236 for 6.
They whacked 48 runs from the final three overs, while Australia, needing 43 in the same time-frame, could manage only 26, despite the best efforts of Marcus Stoinis (56 off 43), as Wood and Woakes kept composed.
Morgan was born in - and even started his worldwide career for - Ireland, having made his debut in 2006.
The Englishman had another scare on 97 when he was given out leg before wicket on the first ball of the last over but the decision was correctly overturned on review after replays showed the ball hit the edge of his bat.
However, the narrative hasn't changed.
"That has probably been our best win as a group to date".More news: I want immigrants to come in from everywhere, says Trump
He was even given a lifeline after Cameron White put down a sitter but failed to make the most of the break.
Mitchell Marsh was the unlucky bowler, but had his revenge a few minutes later when he bowled Ali for six, continuing the all-rounder's wretched tour.
That is when it started to go all horribly wrong for the hosts.
Between them, Buttler and Woakes hammered 11 fours and six sixes and showed just how versatile their batting line-up can be.
Sunday's result was shrouded in controversy, after Steve Smith was given out caught behind trying to drive a Mark Wood (2-46) delivery that went low to wicketkeeper Buttler.
The bucket-load of runs conceded in the death overs on a hard wicket was what ultimately cost Australia the match and the series as they allowed England a total which was at least 30 runs over par.More news: Women's March in Washington against Trump administration
Buttler, for his part, remained adamant he had caught the ball cleanly when queried about it afterwards.