Paul Connor's 32nd minute goal looked to have secured a dramatic victory for the Swans against a side beaten only twice at home all season.
But Ingimarsson thumped a 20-yard shot in off a post with two minutes left to earn Reading a replay at the Vetch Field.
The Icelandic defender's third goal of the season was Reading's first in 360 minutes of football, a barren run which prompted the midweek signing of former England striker Les Ferdinand.
Ferdinand was handed an immediate debut and quickly formed a dangerous looking partnership with Nicky Forster.
Ferdinand might have opened his Reading account from a Nicky Shorey corner after only four minutes but team-mate Ibrahim Sonko got in his way and planted a header wide.
Ferdinand teed Forster up for a wild shot which flew wide and Forster should have done better again when he side-footed across goal after Ferdinand flicked on a Bobby Convey corner.
Convey, making only his fifth start since a £700,000 summer move from DC United, was unlucky to see a stinging shot fly just over as Reading continued to do most of the early pressing.
But Swansea, who had won five of their previous six matches, always looked dangerous on the break and it was not without warning that they took the lead on 32 minutes.
Mercurial midfielder Lee Trundle saw a shot fall kindly into the path of Connor, who squeezed a low effort beyond the reach of goalkeeper Marcus Hahnemann and inside the far post.
Reading laid siege to the Swansea goal in the second half but simply couldn't find a way past a stubborn defence.
Ferdinand, who lasted an hour before making way for Lloyd Owusu, narrowly failed to connect with an Andy Hughes cross in front of goal, and when the impressive Glen Little volleyed over from substitute Dean Morgan's cross it looked as if it was not going to be Reading's day.
But Ingimarsson's ensured Steve Coppell's Royals made it into the fourth round draw with his late salvage strike.
The defender had just seen a tame effort easily cleared following a corner, but when the loose ball fell to him moments later, Ingimarsson let fly with a shot that bounced in off a post.
And Reading were nearly handed a dramatic late winner when goalkeeper Willy Gueret whacked a clearance against the lurking Owusu, and was massively relieved to see the ball bounce just wide.