Ivanovic headed in on 36 minutes after Gary Cahill had helped the ball into his path to give Chelsea the lead from their first and only real chance of the evening at the Parc des Princes.
Edinson Cavani replied with a header of his own in the 54th minute and PSG enjoyed the bulk of the chances but Courtois intervened time and again to keep the Premier League leaders on level terms.
“I think the first half was our half. The second half was their half. I think 1-1 reflects that,” said Chelsea manager Jose Mourinho.
“But if you go to chances created and the goalkeepers’ performances we have to be honest and say they had more than us, so if somebody was closer to winning the game it was Paris not Chelsea.”
Speaking to French television station Canal Plus, Eden Hazard added: “We were winning 1-0 and we could have held on, but we know we are stronger at Stamford Bridge and we will try to qualify there.
“We need to play better and create more chances because we didn’t have many.”
Mourinho’s side should certainly fancy their chances of progressing to the quarter-finals when the second leg comes around in London on March 11, with Paris needing to score an away goal in that game to keep the tie alive.
That was something Laurent Blanc’s side failed to do when the teams met in the last eight a year ago, allowing Chelsea to go through on away goals despite losing the first leg 3-1 in France.
Seven points clear at the top of the Premier League, Chelsea are a stronger side than they were then, while PSG had been forced to deal with something of an injury crisis in the build-up to this game.
With Lucas, Thiago Motta and Yohan Cabaye among the players on the sidelines, Blanc opted to start David Luiz in midfield against his former club while Marquinhos lined up in central defence despite being a doubt ahead of the game.
Chelsea were at full strength but their chief creator, Cesc Fabregas, struggled to make an impact in the match up against the combative Luiz.
In a cagey opening, it was the hosts who created what chances there were, with Blaise Matuidi the first to test Courtois on 11 minutes with a header from a Cavani cross.
The Belgian goalkeeper was equal to that effort and was called into action again seconds later to keep out a Zlatan Ibrahimovic header from a Matuidi centre.
Ibrahimovic and his fellow forwards were afforded little space in the final third and they had to wait until after the half-hour mark to trouble Courtois again, Cavani seeing his near-post header from Ezequiel Lavezzi’s corner turned behind.
- Sucker punch -
Chelsea were happy to soak up the pressure and hit Paris with a sucker punch nine minutes before the interval thanks to a rare attacking combination between three of their back four, Ivanovic stealing in behind Luiz to head home after Cahill had flicked on a John Terry cross.
However, those defenders hardly covered themselves in glory as PSG drew level nine minutes after the restart, leaving Cavani completely unmarked to head home his 17th goal of the season — his sixth in the Champions League — from a Matuidi left-wing cross.
With their backs to the wall, Paris came out fighting in the second half and Chelsea were again grateful to the outstanding Courtois as he saved from Ibrahimovic before Lavezzi’s follow-up was blocked.
Courtois, again preferred to Petr Cech between the posts by Mourinho, then got his fingertips to a Luiz header from a Lavezzi free-kick.
PSG’s unbeaten record at home this season did not appear under threat in the closing stages, but they could not get the winning goal on the night.
Cavani put a glorious chance wide of the far post 10 minutes from time and, when Ibrahimovic got a header on target in stoppage time, that man Courtois was on hand yet again to save Chelsea.