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Wes Hoolahan gave the Republic a deserved lead shortly after half-time with a crisply struck half-volley worthy of opening the scoring in any match.
Martin O'Neill's team had struck the woodwork and gone close several times prior to scoring.
But, after taking the lead they found themselves under increasing pressure and conceded when Clark, chosen ahead of Richard Keogh in the centre of defence, headed a low cross from Zlatan Ibrahimovic into his own net.
The equaliser came from one of the few moments of quality from the Sweden captain, who collected a neat backheel from substitute John Guidetti and raced to the byeline before drilling in a low cross.
Jeff Hendrick drew a late save from Andreas Isaksson but there was to be no winner at the end of a feisty, scrappy match the Republic shaded but could not close out.
Commitment, passion and determination were much in evidence at the Stade de France, with those workmanlike qualities comfortably eclipsing any quality and finesse on show.
But, in the end, neither side had quite enough to prevail and are left to ponder a result that is hardly disastrous but not exactly what either wanted.
Both have remaining Group E fixtures against Italy and Belgium - the two favourites to progress - and have arguably missed out on their best chance to get a crucial victory.
In a tournament with 16 qualifiers from the group stage, three points would have taken one of them a huge step towards reaching the next round.
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The Republic lost all three of their games at Euro 2012, when Italian coach Giovanni Trapattoni set up his team to try to stymie their opponents and snatch the points.
Yet after qualifying from a group that included Poland and Germany, there was no lack of adventure and ambition here in Paris.
Robbie Brady was a menace down the left, Hendrick struck the crossbar with a caressed right-foot strike and drew a superb save with a crisp shot from the edge of the area, and John O'Shea was inches from converting Clark's flick-on.
There was also no disputing the quality of Hoolahan's goal three minutes after the restart.
Seamus Coleman delivered a superb cross after a teasing, jinking run down the right and Hoolahan, who had found space at a premium in a generally congested midfield, superbly smashed home a half-volley from 14 yards across goal and beyond Isaksson.
Hendrick forced another save from the Sweden goalkeeper with a low strike after his team conceded and the men in green showed enough to suggest that whatever happens to them in this tournament they will leave everything on the field.
Ibrahimovic said before the match that "the legend can still deliver", but it did not look like it for most of the match. On many occasions - especially during the first half - the 34-year-old was drawn so deep in search of the ball that he looked more like a midfielder.
Ibrahimovic, strongly linked with Manchester United, scored 11 of his team's 19 goals in qualifying, and going into the fixture with the Republic might have fancied his prospects against a central-defensive pairing of O'Shea and Clark.
Yet he had very little chance to improve on his record of 62 international goals. He struck the wall with a free-kick, snatched a volley and was inches from converting a low drilled cross.
But he was at the heart of the move that led to the equaliser.
The unfortunate Clark had already gone close once to putting the ball in his own net with a header Darren Randolph did well to save before Emil Forsberg hammered the rebound wide.
But Randolph could do nothing to stop Clark's header after Ibrahimovic had finally sprung to life and at least partially justified his top billing.
Republic of Ireland manager Martin O'Neill: "I thought we were magnificent in the game with some really great football.
"Of course the goal deflates you and there is disappointment in the dressing room, but I don't think the players should be downhearted with their performance.
"I think they should be absolutely thrilled with the way we played."
Goalscorer Wes Hoolahan: "The boys were unlucky but we'll take the point. We created a lot of chances.
"In the first half we were excellent and at the start of the second as well. The boys should be proud of themselves, it's a great point in the end."
Sweden take on Italy in Toulouse at 14:00 BST on Friday while the Republic face Belgium in Bordeaux at 14:00 BST the day after.
Match ends, Republic of Ireland 1, Sweden 1.
Second Half ends, Republic of Ireland 1, Sweden 1.
Foul by Zlatan Ibrahimovic (Sweden).
Ciaran Clark (Republic of Ireland) wins a free kick in the defensive half.
Foul by Andreas Granqvist (Sweden).
Shane Long (Republic of Ireland) wins a free kick in the defensive half.
Albin Ekdal (Sweden) wins a free kick in the defensive half.
Foul by James McClean (Republic of Ireland).
Foul by Sebastian Larsson (Sweden).
James McClean (Republic of Ireland) wins a free kick in the defensive half.
Substitution, Sweden. Albin Ekdal replaces Oscar Lewicki.
Substitution, Republic of Ireland. Aiden McGeady replaces James McCarthy.
Guidetti (Sweden) wins a free kick in the defensive half.
Foul by Glenn Whelan (Republic of Ireland).
Offside, Republic of Ireland. Glenn Whelan tries a through ball, but Shane Long is caught offside.
Foul by Zlatan Ibrahimovic (Sweden).
Ciaran Clark (Republic of Ireland) wins a free kick in the defensive half.
Substitution, Republic of Ireland. Robbie Keane replaces Wes Hoolahan.
Glenn Whelan (Republic of Ireland) is shown the yellow card.
Foul by Zlatan Ibrahimovic (Sweden).
Seamus Coleman (Republic of Ireland) wins a free kick in the defensive half.
Andreas Granqvist (Sweden) wins a free kick in the defensive half.
Foul by Ciaran Clark (Republic of Ireland).
Corner,  Republic of Ireland. Conceded by Martin Olsson.
Attempt missed. Shane Long (Republic of Ireland) header from the centre of the box misses to the left. Assisted by James McCarthy with a cross.
Foul by Kim Källström (Sweden).
Wes Hoolahan (Republic of Ireland) wins a free kick in the attacking half.
Attempt saved. Jeff Hendrick (Republic of Ireland) right footed shot from a difficult angle on the right is saved in the bottom right corner. Assisted by Glenn Whelan.
Own Goal by Ciaran Clark, Republic of Ireland.  Republic of Ireland 1, Sweden 1.
Attempt missed. James McClean (Republic of Ireland) header from the centre of the box misses to the left. Assisted by Wes Hoolahan following a set piece situation.
Foul by Andreas Granqvist (Sweden).
Wes Hoolahan (Republic of Ireland) wins a free kick on the right wing.
Hand ball by Wes Hoolahan (Republic of Ireland).
Offside, Sweden. Martin Olsson tries a through ball, but Zlatan Ibrahimovic is caught offside.
Offside, Sweden. Martin Olsson tries a through ball, but Zlatan Ibrahimovic is caught offside.
Substitution, Republic of Ireland. James McClean replaces Jonathan Walters.
Offside, Republic of Ireland. Ciaran Clark tries a through ball, but Jonathan Walters is caught offside.
Offside, Sweden. Guidetti tries a through ball, but Zlatan Ibrahimovic is caught offside.
Nilsson-Lindelöf (Sweden) is shown the yellow card for a bad foul.
Foul by Nilsson-Lindelöf (Sweden).
Ciaran Clark scored an agonising own goal as the Republic of Ireland were held to a frustrating draw against Sweden in their opening Euro 2016 fixture in Paris.