The home side went a man and a goal down three minutes before the break when Hawsawi was dismissed for handling the ball on the goalline and Ali Karimi stepped up to convert the resulting penalty.
But a brilliant free-kick by Mohammed Al Shalhoub early in the second half restored parity for the home side at the Prince Faisal bin Fahd Stadium.
With the earlier match between Al Shabab Al Arabi of the United Arab Emirates and Al Gharafa of Qatar finishing goalless, all four teams in Group D now have one point after the opening round of games.
In an opening half of few real chances, Al Hilal's Ahmed Al Fraidi nearly broke the deadlock in the 34th minute with a curling shot that flew narrowly wide of the right post.
Piroozi created little early on but the match turned in their favour when Hawsawi stretched out his hand while lying on the ground to keep out a shot by Mohammad Nouri that was heading into the bottom corner of the net.
He was shown the red card and Ali Karimi calmly chipped the ball past goalkeeper Khalid Shrahili to give the Iranian side the lead.
Adil Hermach wasted a chance to restore parity for the home side before half-time when his free-kick sailed over the crossbar.
But Al Shalhoub would make no mistake eight minutes after the break when he bent a free-kick from 25 metres into the top left corner of the net, giving goalkeeper Alireza Haghighi absolutely no chance.
Yoo Byung-soo had a great opportunity to put the 10 men of Al HIlal ahead shortly afterwards but the Korean stabbed his effort wide.
Karimi then had a big opportunity to restore Piroozi's lead with 20 minutes to go with a free-kick taken from a similar position to Al Shalhoub's but the Reds skipper saw his effort into the top left corner tipped away by Sharahili.
As time ticked to the final whistle, Youssef El Arabi charged past two Piroozi defenders on the left byline but saw his effort from a narrow angle tipped away by Haghighi.
Despite seeing his side fail to hold on to their lead despite the advantage of an extra man, Piroozi coach Mustafa Denizli was satisfied with the result.
"The strongest two teams in the competition played against each other tonight and I think that it was a good match," said the Turk.
"We had several chances to score but it is not always easy when you are playing against a team with 10 players.
"It will be important for us to win the return game against Al Hilal in because we have huge support there. To progress to the next round, we must not lose any of our away games and win at home."
Al Hilal coach Ivan Hasek felt that his team had done their best in the circumstances of the match.
"We tried to win and make the most of our chances. However, it was not an easy task especially because Piroozi have a strong defence," said the Czech.