With both openers off key, Root and Morgan's partnership was vital for the hosts.
Liam Plunkett was the only England bowler to go wicketless in the match.
England come into the fixture in good form, having beaten their visitors by seven wickets in Bristol on Friday.
Ireland had England a little on the ropes at that stage but Bairstow (72 not out) and Adil Rashid (39) took England past the 300 mark.
Morgan, who played some handsome 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.
Joe Root's shined with the bat, scoring 73 runs, and also the ball, picking up three wickets to drive England home to an 85 run win in the second ODI against Ireland and the win the two-match series.
Stirling carved eight fours and a six from 42 balls to give the Irish reply a flying start, and reason to believe.
The game wasn't much of a pointer towards England's Champions Trophy ambitions, but it could prove to be a vital springboard for Rashid.
The Yorkshireman has been one of his country's most consistent batsmen in Test cricket over the past 18 months.
However, they played with greater freedom and application on Sunday, rising to the challenge of their first encounter against England at Lord's.
England exercised caution through the middle overs - in the Indian Premier League-enforced absence here of three of the players who give their first-choice one-day worldwide line-up such envied power and depth.
The lack of depth is mirrored in their batting.
'Sterner tests to come' - What they said..
But no contribution was more eye-catching than that of Bairstow, who hit ten boundaries during a stunning innings that may now force coach Trevor Bayliss into a rethink ahead of next month's Champions Trophy, a tournament England are hosting as favourites. "For us to step on our game from the start was something to focus on leading into the South Africa series and the Champions Trophy". But he was back to his best with the white ball as Morgan's men opened their high-profile summer by charging to an early victory to go 1-0 up with one more match to play in the Royal London Series, tomorrow at Lord's.
"It was Adil's day today", said Morgan, England's Dublin-born ODI captain.
"There was some cloud cover this morning and the openers got us off to a fantastic start". I want to be a part of every England side that there is going forward because, with the group of players that we've got, it's definitely an exciting blend of players.
"It wasn't the seamers that damaged us, it was the spin", Porterfield said. "If we'd had those couple of wickets in hand coming down to the back end, to the last 15, I think it could have been a very different game".
Alex Hales took advantage of two reprieves with a 33-ball fifty and the main challenge for England was whether they would complete victory before the interval was needed.
Willey had proved expensive but struck back when Ed Joyce fell across a straight delivery that would have taken leg stump and the two quick wickets forced Ireland to take stock. "You're looking at someone of his quality - and Stokes, Woakes, Buttler that are going to come into your first XI".
"I couldn't have asked any more from the boys today".
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