Leg-spinner Rashid took five for 27 as Ireland, in their long-awaited first ODI against England in England, collapsed to 126 all out after visiting captain William Porterfield won the toss.
Scores in excess of 300 are routine in ODIs these days and Jonny Bairstow ensured the home side achieved that with three sixes in his 72 not out from 44 balls but Chase and McCarthy can be satisfied to have conceded fewer than seven runs per over.
Ireland, however, will have the chance to make quick amends for Friday's reversal - its first fixture against England in England - when it concludes the two-match series with a maiden global at Lord's, the "Home of Cricket", in northwest London, on Sunday.
They were energetic in the field but unfortunately lack bowling depth beyond Tim Murtagh, who used his experience of Lord's from 10 years as a Middlesex player to keep England in check, especially in an initial spell of 1-16 off his opening six overs. In the 29th over of the innings, Root got to his fifty off 57 balls and a ball later, Morgan raised his fifty off 49 balls.
Morgan, who played some attractive shots, was fortunate to survive when the ball rolled onto the stumps in the 23rd over off the bowling of Paul Stirling when he was on 40, but the ball didn't dislodge the bails.
Porterfield could not find any fluency on his return to the ground where he started his professional career, and eventually fell in anti-climax for 13 off 45 balls when he presented new England Test captain Root with a wicket from only his second delivery of the new worldwide season.
Morgan went on to make 76 off 79 deliveries, which included five boundaries and two sixes, before he was sent packing by Barry McCarthy.
After Friday's first ODI, England captain Eoin Morgan spoke of the need for his players to find form before June's Champions Trophy on home soil, for which England are now bookmakers' favourites.
The shadows of the Grandstand were slowly making their way towards the the square as Ireland's tailenders pushed their reply to 243 after England had posted a total of 328 for six having been put into bat.
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Joyce fell 15 balls later, bowled by Root, and by the time Andy Balbirnie was trapped lbw by Liam Plunkett Ireland were 87 for three in the 19th over.
"It was Adil's day today", said Morgan, England's Dublin-born ODI captain.
But their recent 7-2 reverse across three formats against fellow Test aspirants Afghanistan in India was an indication of how they are struggling to match the standards of previous Ireland teams that beat major nations at several World Cups.
Morgan added: "We should have been more clinical, one of us should have got a big score, but Jonny and Adil led us home really well".
"I thought we bowled very well and could quite easily have restricted them to maybe 20 less". Hopefully we're 2-0 up with one to go and we'll have a look at them then.
At the other end, Peter Chase was as much wayward as Murtagh was controlled.
"That's something we need to mentally put right for Sunday".
"Not taking anything away from Rashid, we should have played it a lot better".
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