The Sweden captain sunk the plucky Faroes on 75 minutes, flicking Pontus Wernbloom's delicate first-time pass beyond keeper Gunnar Nielsen before firing home.
"That's a huge relief. Three points are vital if we are to be fighting to make the finals in Brazil," relieved Sweden coach Erik Hamren told TV4.
Hamren's team now have two wins from two games in the group. The Faroes have lost both their matches.
The fired-up Faroes, playing on their new artificial pitch for the first time, stunned the Swedes on 57 minutes when a weak Jonas Olsson clearance fell to Rogvi Baldvinsson and he scored with a low shot past Andreas Isaksson.
Sweden coach Erik Hamren made an inspired double substitution just after the hour, bringing on Anders Svensson and Alexander Kacaniklic.
The move quickly paid off when Ibrahimovic, limited to speculative shots from distance in the opening half, linked up with Svensson and set up Kacaniklic for the equaliser in the 65th minute.
Ibrahimovic was less than pleased with the new surface.
"It caused a lot of problems... for the guy on the ball and the guy receiving it," he said.
"But the most important thing is that we won the game... these games are not easy."
