The Uruguayan striker fired home a 71st-minute penalty to maintain PSG's unbeaten record after 16 games despite a disjointed performance without injured star attacker Neymar, equalising Kenny Lala's own spot-kick four minutes before the break for the spirited hosts, who came close to inflicting their opponents first league defeat of the season.
"If you arrive in flip flops against PSG, you go home with a good spanking," said Strasbourg coach Thierry Laurey.
"We suffered at times but we managed to minimise their offensive power. They dominated the play but we were never in too much danger."
Thomas Tuchel's side, who also drew at Bordeaux at the weekend and started without Kylian Mbappe and Thiago Silva, remain way ahead of the rest after a night in France full of late goals, incident and VAR.
Champions PSG are 14 points ahead of Lille, who moved second on Tuesday after beating Montpellier 1-0 and stayed there thanks to Hatem Ben Arfa and Jordan Siebatcheu leading Rennes to a 2-0 win at Lyon on Wednesday.
Bruno Genesio's Lyon are fourth, 15 points behind PSG, while inconsistent Marseille are a further two points back after a 3-2 defeat at Nantes.
PSG now don't play until they travel to Serbia for their Champions League clash at Red Star Belgrade, where a win will ensure their progress to the last 16, after Saturday's match with Montpellier was postponed because of security concerns.
"We reacted and showed a lot of quality in the second half. We completely controlled play in the opposing half and had both a lot of possession and dangerous situations," said Tuchel.
"We played very well in the second half, showed quality but could not score."
Night of drama
The video assistant referees were involved in a series of key decisions on Wednesday, and it was VAR that helped Strasbourg take the lead over PSG, with referee Antony Gautier awarding Lala's spot kick a full four minutes after Thilo Kehrer handled Lionel Carole's fizzing cross.
The half-time deficit led Tuchel to bring Mbappe off the bench, and it was France's World Cup star who won the penalty that Cavani blasted past Matz Sels, Lala hacking the 19-year-old superstar down in the box.
PSG looked a different proposition to the sluggish team that slogged through the opening period, and Kehrer thought he had made amends for giving away the opening penalty six minutes after the break, only to see his header saved at point-blank range by Sels.
However as the half wore on the away side ran out of ideas and Strasbourg made a push for the win in the final minutes.
Substitute Adrien Thomasson forced a superb stop from Alphonse Areola with a minute left, and the 24-year-old thought he had snatched a famous win deep in injury time when he headed home Youssouf Fofana's pinpoint cross.
However while the majority of the crowd celebrated wildly, the linesman's flag was rightly raised to rule the unfortunate Thomasson's goal out for offside.
Chief among the other crucials decisions helped by VAR on a dramatic night was Loick Landre's 96th-minute winner for Nimes at Caen, which came after Prince Oniangue had levelled at 1-1 in the 90th minute and was eventually given after Landre was found to be onside.
VAR also came to Bordeaux's assistance by awarding the penalty slotted home by Francois Kamano as they followed up on drawing with PSG at the weekend with a last gasp 3-2 win over Champions League-chasing Saint-Etienne at the Matmut Atlantique.
Pablo bagged the points in the final minute, and Bordeaux's opening goal came at a cost to Neven Subotic, who was stretchered off with blood pouring down his face after being clattered by his own goalkeeper Stephane Ruffier just before Jimmy Briand scuffed in.
Dijon sank rock-bottom Guingamp further into the relegation mire thanks to Oussama Haddadi scoring four minutes in time to hand the hosts a 2-1 win.