Napoli play a late home game against Genoa late Saturday. Napoli is 12 points behind Inter.
Inter was in need of a boost after letting slip a two-goal lead to lose 3-2 at Borussia Dortmund in the Champions League on Tuesday.
"I saw an important response after Dortmund", said Conte.
Inter gave away a penalty in the 19th minute when goalkeeper Samir Handanovic caught Mattia Zaccagni as he raced through and Valerio Verre made no mistake from the spot.
But Barella grabbed the victor with a superb curling effort on 83 minutes whipping off his jersey as he celebrated wildly with his teammates.
It was a first win in seven matches for Torino.
"Now it's time for the national team and I'm already focused on that", he added of Italy's upcoming Euro 2020 qualifiers.
In the earlier game, Brescia's new coach Fabio Grosso had a nightmare debut as his side were thumped 4-0 at home by Torino.
The second came on 26 minutes when Czech defender Ales Mateju was also deemed to have used his arm to block a Belotti shot after a VAR review.
Barella collected the ball on the left, cut inside and curled the ball into the far corner from the edge of the area for his first goal Serie A goal since joining Inter from Cagliari at the start of this season.
Brescia were reduced to ten men before the break when Mateju was sent off for a second yellow card for fouling Belotti.
A man down, Grosso substituted Brescia forward Mario Balotelli by defender Bruno Martella after the break.
Jesse Joronen denied Belotti a hat-trick before Berenguer fired past the Finnish keeper with 15 minutes remaining.
Inter scored two second half goals to cancel out Verona's first half lead to take three points and move to the top of the league table.
Torino, who had clinched two points from the past six games, are now 11th on 14 points from 12 games.