Victory saw England go 1-0 up in the two-match contest ahead of Sunday's showpiece clash at Lord's.
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.
Andrew Balbirnie top-scored with 30 in an innings where only openers Ed Joyce (23) and Paul Stirling (20) got to 20.
Rashid's return was second only to that of off-spinner Vic Marks's five for 20, against New Zealand at Wellington in 1984, by a specialist England slow bowler at this level.
Fellow opener Alex Hales was then dropped on nought, Middlesex quick Tim Murtagh failing to hold a tough caught and bowled chance.
Hales went to make a quickfire 55 that stopped any thoughts of a stunning upset in their tracks.
Both Hales and Dublin-born England captain Eoin Morgan holed out off Chase, the only Ireland bowler to take wickets in a return of three for 44.
But there was never any chance of England losing, with Test captain Joe Root finishing on 49 not out.
"This is our first game of a long summer. Hopefully we have shown we can do some damage and we can carry it on," the man-of-the-match added after surpassing his previous ODI best of four for 43 against Bangladesh in Chittagong in October.
- 'Leg-spin threat' -
=====================
Meanwhile Morgan was pleased to see Rashid back in the wickets after a "tough winter" in India.
"Adil's day shows the threat leg-spin poses to any team," Morgan told Sky Sports.
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 Irish teams that beat major nations at several World Cups.
Porterfield said Thursday that teenage leg-spinner Rashid Khan had been the difference in their defeats by Afghanistan.
Fellow leg-spinner Rashid caused Ireland fresh embarrassment as several batsmen played poor shots against the Yorkshireman.
"It wasn't the seamers that damaged us, it was the spin," Porterfield said.
"We could have played that a lot better."
English county batsmen Joyce (Sussex) and Stirling (Middlesex) made a solid start.
Stirling struck five fours before his middle stump was uprooted by fast bowler Mark Wood, making his first England appearance since September after a recurrence of an ankle injury.
Ireland were 40 for one and soon afterwards Joyce was lbw to David Willey.
A third-wicket stand of 35 between Porterfield and Balbirnie followed but there was no stopping the collapse once it started.
Warwickshire batsman Porterfield's sluggish 13 off 45 balls ended when he chipped part-time off-spinner Root's second ball to mid-off.
Six years ago Kevin O'Brien rescued Ireland from a similarly dire position of 111 for five with the fastest ever World Cup hundred to set up a memorable win over England in Bangalore.
But he has not hit those heights since and on Friday he was lbw for four, failing to pick Rashid's googly.
And when Rashid had Murtagh caught in the deep, Ireland were all out in a mere 33 overs.
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