Ireland deservedly led through Wes Hoolahan early in the second half after Jeff Hendrick rattled the crossbar during the opening period.
Ibrahimovic was well shackled by a diligent Irish defence for the most part but surged beyond John O'Shea in the 71st minute and smashed the ball across goal for the unfortunate Ciaran Clark to turn into his own net.
Earlier, Clark haphazardly turned a corner towards the Ireland goal, with goalkeeper Darren Randolph sparing him on that occasion.
The Aston Villa centre-back tussled manfully with Ibrahimovic throughout the Group E match but his ever-assured opponent teased that sorcery might have been at play.
"How do I make defenders score? I gave the cross and he headed it into the goal," Ibrahimovic told reporters.
"Was it magic? Maybe you should ask him."
Ireland's players were understandably unwilling to join Ibrahimovic in making merry at Clark's expense.
"These things happen," said forward Jon Walters, who left the action early due to an Achilles injury.
"I think if [Clark] doesn't get a touch then [Sebastian] Larsson behind him puts it in.
"Ciaran's done really well all game. The two centre halves kept Zlatan at bay the whole game.
"Then he makes a couple of unbelievable clearances after [the own goal]. So, you know, these things happen."
Everton full-back Seamus Coleman, who brilliantly created Hoolahan's opener, added: "Clarky is very experienced, very mature. He knows that no one is pointing the finger or anything like that."
There was just Ibrahimovic, playfully pointing the magic wand.