The 21-year-old became England's youngest captain in any format after winning the toss on a sunny day in Malahide, opting to bowl first on a pitch that proved a batsman's paradise.
Salt's fireworks display featured 10 boundaries and four sixes from just 46 balls, though he fell agonisingly short of what would have been a second consecutive century after his unbeaten 141 against South Africa last week.
"I enjoyed that, it's a pretty good wicket and a good ground to bat on when you get going. It was fun," Salt said.
"There was no chat in the dressing room other than to bat, but it was a really good effort from Ireland, they did really well to get that total."
Earlier, Ireland had posted a challenging total thanks to a blistering 123-run partnership between Harry Tector (61 not out) and Lorcan Tucker (55) that came off just 68 deliveries, after openers Paul Stirling and Ross Adair had provided the platform.
Salt and Jos Buttler (28 off 10) then responded with a rapid 74-run opening stand, while Bethell chipped in with 24.
Although England lost six wickets, the tourists cruised home with 14 balls to spare to take a 1-0 lead in the three-match series.
"It feels pretty special. Winning the first one is always nice," Bethell said.
"It was a good total. A ground like this was a bit of an unknown. Coming off we thought it was maybe a couple too many. Going into Friday and Sunday, we know the ground more, the dimensions and how to take wickets here."
