Wes Hoolahan's goal sparked scenes of joy amongst the Irish supporters at the Stade de France and it looked like Martin O'Neill's men would pick up their first points at a European Championship finals in 28 years and their first ever competitive win over Sweden.
But Sweden heaped pressure on the Irish goal and Clark suffered the anguish of costing his side two vital points when he headed Zlatan Ibrahimovic's cross into his own net.
Ireland worked tirelessly to try and take three points from their easiest-looking Group E fixture and deny Ibrahimovic the chance to make his mark on the tournament, but a tenacious first-half performance gave way to a tired final quarter and Sweden capitalised.
Ibrahimovic was well shackled for much of the game as he sought a goal in a record fourth European Championship and Cristiano Ronaldo can now beat him to the honour of that particular slice of tournament history when Portugal face Iceland on Tuesday.
The men in green shook off their early nerves and Jeff Hendrick drew a good save from Andreas Isaksson in the Sweden goal with a smart shot on the half-volley inside the opening 10 minutes.
Clark flashed a header across the face of goal that was inches away from John O'Shea as the defender slid in but the chance went begging as Ireland began to control the game.
O'Neill's side successfully pressed and harried the Swedes, whose chances were restricted to a long range free-kick that Ibrahimovic slammed against the Ireland wall.
At the other end, Ireland's best play centred around Hendrick, who teed up Robbie Brady for a shot the flashed just over the crossbar, before the Derby County man crashed a dipping effort of his own onto the woodwork from 25 yards.
But Ireland took a deserved lead three minutes after the interval. Seamus Coleman jinked inside from the right flank before sending over an out-swinging cross that Hoolahan powered into the net on the half-volley.
The goal sparked Sweden into life, and Clark had to hook a shot over the crossbar before Emil Forsberg squandered a simple chance on the rebound from a fine Darren Randolph save.
Ibrahimovic turned a close-range shot just wide of Randolph's goal as Sweden continued to press for an equaliser and with 20 minutes left the Irish resolve was broken.
Sweden swept into the Ireland penalty area and Ibrahimovic reached the by-line before squaring a cross that Clark headed into his own net, with Sebastian Larsson waiting to convert behind him.
Marcus Olsson flashed a cross just past Ibrahimovic as Sweden swarmed forward, but both sides were left with a point that likely leaves them requiring stellar performances against group favourites Belgium and Italy to progress.
