England, having lost to New Zealand by 12 runs last week, need to win the match and finish with a better overall run-rate than the Kiwis if they are to make Wednesdays final in Auckland against Australia.
Morgan, who missed the last two England matches because of a groin strain, had six sixes and four fours in his crucial innings.
He went to the middle in the fourth over when England were two for 24 and featured in a 93-run, nine over stand with Dawid Malan.
With a place in the finals riding on the game, New Zealand skipper Kane Williamson did not want to let the England batsmen settle when he won the toss and opted to bowl.
He mixed up his attack, using seven bowlers in the first 10 overs as England reached the halfway stage at two for 85.
Although Williamsons strategy had seen an abundance of dot balls, England did have six sixes and five fours to their credit.
Malan and Morgan ramped up the pace in the 11th over when they belted 21 off an over by Mitch Santner on the way to England scoring 109 in the second half of their innings.
He had the wicket of Alex Hales for one and Boult switched ends to claim Roy for 21.
Williamson tried himself against the left handers Malan and Morgan and although he went for a costly 16 he stayed with the spin option and brought Ish Sodhi and Colin Munro.
Morgan, however, was able to score freely while no one else managed more than 10 once Malan departed.
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