Playing at the ground he grew up on, Dublin-born Morgan combined with Bopara for an ODI record fifth-wicket stand of 226 in 169 balls.
They had come together with England in trouble at 48 for four, after Ireland captain Will Porterfield's century helped the hosts post 269 for seven.
But Morgan, who had not passed 50 in his previous 19 ODIs, took charge alongside Bopara to ensure they will be in peak form ahead of the five-game series against Australia.
Morgan offered praise for Ireland who saw another of their former number, Boyd Rankin, take four wickets in his ODI debut for England.
A record sell-out crowd of 10,000 watched on at Irish cricket's new headquarters, which were showcased on the international stage for the first time.