Gary Wilson struck 80 from 69 balls in a late-innings burst to help the Irish edge the United Arab Emirates by two wickets with four balls to spare.
In the unfamiliar position of being the favourite after its opening upset win over the West Indies, Ireland won the toss, sent the UAE in to bat and had them in trouble at 78-4 before Shaiman Anwar came to the rescue with 106 to lift the total to 278-9.
The Irish never appeared comfortable in reply after losing Paul Stirling (3) in the second over, eventually sneaking over the line with key contributions from Wilson and Kevin O'Brien (50).
Needing almost 10 runs an over at one stage, Wilson and O'Brien smashed 72 runs in their sixth-wicket stand off just six overs to revive the chase.
"Closer than we'd have wanted, but we'll take a win," Ireland captain William Porterfield said. "It helped to have wickets in hand, especially when you have Gary Wilson and (O'Brien) coming in at the end."
Mohammad Tauqir praised the contributions of Anwar and Amjad Javed, who scored 42 in a vital partnership and returned to take three wickets and a crucial catch.
"Shaiman played a superb knock, to go from 130-6 to 280, the credit goes to him, brilliant innings," Tauqir said. "Both of them played superbly.
"We bowled well up to 35 overs, and later on there was dew and the ball became wet — and it was a very good batting wicket and they batted well. It went to the wire."
For the first time in the tournament, there will be two matches on a mid-week day. On Thursday, Afghanistan plays Scotland at Dunedin, New Zealand, with the winner set to pick up its first-ever win at a World Cup and Sri Lanka plays Bangladesh at the Melbourne Cricket Ground.
Sri Lanka opened with a 98-run loss to New Zealand and beat Afghanistan by four wickets with 10 balls remaining.
Both teams have had personnel issues in the past few days, with Sri Lankan allrounder Jeevan Mendis ruled out because of a hamstring injury he sustained at training on Tuesday. Upul Tharanga was flying to Melbourne to take his place in the squad.
Bangladesh suspended pace bowler Al-Amin Hossain for breaking a team curfew in Brisbane.
Sri Lanka has won 32 wins of 37 one-day internationals against Bangladesh.
Afghanistan is playing at its first World Cup in the 50-over format and Scotland is at its third. No team has faced greater odds of reaching the tournament than Afghanistan, which has been a member of the International Cricket Council for only 14 years and has a team made up in part by players who learned the game in refugee camps while in exile from their war-torn homeland.
In Auckland, the hype continues to build for one of the most-anticipated matches in the tournament — unbeaten New Zealand playing Australia in a game that could decide which team finishes atop Pool A — on Saturday.
New Zealand allrounder Corey Anderson said he expects Australia to use sledging or verbal intimidation to try to unsettle his team, which has won its first three matches. And Australian opener David Warner, never one to mince words, said he expects to be the target of abuse from the sellout crowd at Eden Park which will watch the first ODI between the teams in four years.
"I hope they come out and boo us," Warner said of the Auckland crowd. "That's what's going to happen. We love it, it gets us up and going, gets the adrenaline going."
Meanwhile, Zimbabwe fast bowler Tendai Chatara was reprimanded after pleading guilty to "conduct contrary to the spirit of the game" during his team's 73-run loss under the Duckworth-Lewis system to the West Indies on Tuesday.
The incident happened in the 50th over when Chatara bowled consecutive high full-tosses to Chris Gayle, who scored a World Cup-record 215 containing 16 sixes.Suggest a correction