The hosts" Calum MacLeod said: "To beat a full England side is a massive statement from Scottish cricket.
However, England were undermined by a succession of soft dismissals after Bairstow holed out in the 18th over.
Calum Macleod scored an unbeaten 140 off just 94 balls as Scotland registered their highest score in ODIs, ending at 371/5 in their 50 overs after being put into bat by England.Macleod's 140* was his 4th highest ODI score, earlier he has registered scores of 175 (vs Canada), 157* (vs Afghanistan) and 154 (vs Papua New Guinea).More news: IHOP reveals name change to International House of... Burgers?
Moeen picked out long-on and Adil Rashid was needlessly run out at the start of the 49th over, before Safyaan Sharif trapped Mark Wood lbw with a superb, inswinging yorker to end England's innings on 365.
"A one-off game like this, to bring your best game is very hard to do, they played well and deserved the result", said Morgan.
Australia spinner Ashton Agar, who will face England when the series gets underway at the Oval on Wednesday, said he was happy the controversial opener is in the right head space again following the huge fall-out from the incident in Johannesburg in March.
"We've had a professional relationship with Dave for five years now we've always found him to be a pretty normal and humble bloke - he just wants to atone for his mistakes and move on".More news: Larry Kudlow In The Hospital After Having Heart Attack
"This does not take away what happened in Zimbabwe and everyone's feelings about the ten team World Cup but this was a special day".
Here, Press Association Sport takes a look at the reaction on social media to Scotland's stunning win. That is the bigger shop window rather than personal side of it. 'We were miles off where we expect to be and didn't do enough.
"If ICC and other global teams look at this game they will see we are not an add on". We didn't do enough to win.More news: After 'difficult' G7 summit, May warns against unilateral action on trade