Taylor, brought back after England rested Ashes-winner Joe Root from this series, was out for 101 having been left short of a hundred on 98 not out when England were beaten by eventual champions Australia at the World Cup in Melbourne in February.
The diminutive 25-year-old Nottinghamshire batsman, playing his 21st match at this level, faced 114 balls.
But it was a testament to Australia's accuracy that he only managed five fours.
And whether England, who at 2-0 down with three to play against world champions Australia after a 64-run loss at Lord's on Saturday had to win this match to keep the series alive, had enough runs on the board remained to be seen.
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But the partnership would have been curtailed had not wicket-keeper Matthew Wade missed a stumping off ODI debutant spinner Ashton Agar when left-hander Morgan was on 15.
As it was Australia slowed England's run-rate, with a total of 200 for two in 32 overs, advancing only to 245 for four in the 43rd over.
After Taylor holed out in the last over, it needed some clean-hitting from Chris Woakes, including a six off Mitchell Starc, to take England to 300.