Hoolahan met Seamus Coleman’s flighted ball with a controlled first-time strike three minutes into the second half at the Stade de France to give Ireland a lead they fully deserved.
However, having been kept quiet for long spells by the Irish defence, Sweden captain Zlatan Ibrahimovic created the equaliser in the 71st minute when his cutback was turned into his own net by Aston Villa defender Clark.
It is a result that keeps both sides in the running in Group E, which also contains Italy and much-fancied Belgium, but the Irish will be especially frustrated at leaving with only a point.
“It was a fantastic set up by Seamus Coleman, I caught it on the half volley and was delighted,” Hoolahan told the BBC.
“To get a goal for Ireland in the Euros is a great achievement. 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 half as well. The boys should be proud of themselves.”
Ibrahimovic was attempting to become the first player to score in four separate European Championships, but the Sweden captain was a peripheral figure in a first period in which Ireland were the better team.
Martin O’Neill’s men played with far greater intensity and were first to threaten from a Jeff Hendrick shot nine minutes in that was well struck but was turned around the post by Andreas Isaksson.
Then Clark headed on a left-wing corner and veteran captain John O’Shea, stretching at the far post, was just unable to turn the ball in with the goalkeeper beaten.
– Hendrick hits bar –
Robbie Brady, an attacking player at club level but deployed as a left-back for Ireland, crashed a rising right-foot shot inches over, and in the 32nd minute Hendrick played a one-two with Shane Long before curling an effort from 25 yards onto the bar.
Meanwhile, former Paris Saint-Germain star Ibrahimovic was unable to trouble Irish goalkeeper Darren Randolph from a long-range free-kick.
Sweden were dealt a blow right on half-time when right-back Mikael Lustig came off with what appeared to be a groin injury.
Coach Erik Hamren was forced into a defensive reshuffle as substitute Erik Johansson came on in central defence and Victor Lindeloef moved out to the right.
That defence was finally breached three minutes into the second period. Coleman did superbly on the right and lifted the ball towards the far post where Hoolahan, all alone, smashed home a delightful first-time shot beyond Isaksson.
The sea of Irish fans behind the goal, bedecked in green, erupted, but having created almost nothing themselves, Sweden were nearly level barely a minute later.
Clark turned a corner towards his own goal but Randolph made a superb save, diving low to his right to scoop the ball away.
Ibrahimovic finally came to life on the hour, firing just wide with his back to goal following a Martin Olsson centre. He had a new partner in attack by that point, with John Guidetti having replaced Marcus Berg in a switch which proved pivotal as Sweden dug out the equaliser.
Guidetti’s flick released Ibrahimovic, and his ball from the left was headed past Randolph by the unfortunate Clark.
Ibrahimovic then came within a whisker of converting a low driven cross by Olsson, but a point was the very least Ireland deserved as the match ended all square.