PSG, expected to ease to the title, were missing new signings Zlatan Ibrahimovic and Thiago Silva, who are not fully fit, but still had the best chances before Argentine Lavezzi was dismissed on the hour for a two-footed jumping tackle.
Jeremy Menez had a gilt-edged opportunity but his effort was saved amid PSG pressure while Eduardo hit the post with a classy strike for the Corsicans, relegation contenders last term.
With the game already into injury time, the floodlights at the modest Francois-Coty stadium suddenly went out and fans and players stood about wondering what was going on as the referee chatted with Ajaccio officials.
Eventually the lights came back on and the teams played out the remaining few minutes with no incidents.
PSG coach Carlo Ancelotti, sent to the stands before the blackout for kicking a bottle in disgust, also saw his side draw their opener last weekend when ex-Milan forward Ibrahimovic spared their blushes with a late leveller at home to Lorient.
Olympique Marseille and Andre-Pierre Gignac endured a torrid time last season but the burly front-man finally came good with the first goal in a 2-0 win over Sochaux at the Stade Velodrome.
Rod Fanni sealed the win late on as the southerners racked up a second straight league win under new coach Elie Baup.
"It's good for our confidence. It's a good start to the season. It has been a while since OM started a season like this," Gignac told reporters after scoring following a 54th minute corner and having other chances.
Girondins Bordeaux, the 2009 champions who were just as anonymous as 2010 winners Marseille last term, also made it a maximum six points when Poland midfielder Ludovic Obraniak netted his second of the season in a 1-0 win over Stade Rennes.
Henri Saivet missed a penalty midway though the first half but Obraniak fired in 15 minutes from time for the hosts.
On Saturday, Marseille's next opponents Montpellier again stuttered in their title defence with a last-gasp 2-1 loss at Lorient.
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