Vidic, signed from Manchester United, conceded a first-half penalty that Samir Handanovic kept out before he was given his marching orders in the 92nd minute for dissent.
Handanovic saved Marcelo Larrondo's poor spot-kick in Sunday's clash in what was a lacklustre display from Walter Mazzarri's men.
Torino, in contrast, will rue what could have been after Fabio Quagliarella wasted a golden opportunity to end the club's 20-year wait for a win over Inter, but he dragged his shot wide.
Inter had chances of their own, with substitute Pablo Daniel Osvaldo denied by the impressive Daniele Padelli on a frustrating night for Mazzarri's team.
Vidic was one of three close-season signings handed their first Serie A starts by Mazzarri, who watched as his side dominated possession in the opening exchanges.
Yann M'Vila nearly opened his Inter account after just two minutes, drilling wide from the edge of the area after Torino failed to clear their lines.
There was little sign of a repeat of last season's stunning 3-3 draw in the corresponding fixture during a cagey opening 15 minutes, with the hosts content to pile men behind the ball before attempting to supply new signings Quagliarella and Antonio Nocerino on the break.
That gameplan was nearly rewarded after 20 minutes, when Quagliarella was brought down in the area by Vidic, only for Handanovic to parry away Larrondo's resulting penalty.
Torino, who were hit by the news of Alessio Cerci's derparture to Atletico Madrid before the game, continued to look dangerous, despite not having as much of the ball as their opponents.
A lively Quagliarella should have hit the target after 32 minutes, powering a header over following a well-worked corner, before Inter's Mauro Icardi steered a teasing Dodo free-kick well wide.
But that was as close as either side came during a closely-fought first half, with neither goalkeeper really tested.
That changed seven minutes after the break, when Cesare Bovo's curling 25-yard strike was smartly saved by Handanovic.
Handanovic breathed a sigh of relief just a minute later, when some good work from Matteo Darmian saw Larrondo set up Quagliarella, who poked an awkward shot just wide from close range.
Mazzarri threw on another new signing in Osvaldo in an attempt to settle his side and the Italy international nearly found a breakthrough after 68 minutes.
A defence-splitting pass from Hernanes saw Osvaldo clean through on goal, only to be expertly thwarted by goalkeeper Padelli inside the area.
Having already conceded a penalty, Vidic's debut to forget was compounded when he was shown a straight red card for sarcastically applauding referee Daniele Doveri deep into stoppage time.