The Serie A runners-up have won four of their last five league games. PHOTO: AFP
Gattuso will make his Champions League debut as a coach against the Spanish, having replaced Carlo Ancelotti, who was sacked despite leading the southerners to the knockout rounds.
"Don't make me think, leave me alone," urged Gattuso after the win which moved his side into the Europa League places.
Lorenzo Insigne and Fabian Ruiz's second-half goals rescued a win after Jhon Chancellor had put the northeners ahead after half an hour.
"We will think about (Barcelona) tomorrow, today only Brescia counted.
"It's a competition I'll lose sleep over, but on the other hand I almost never sleep now," continued the 42-year-old, a former World Cup winner who won two Champions League titles as a player with AC Milan
"When I talk to my colleagues who are a few years older than me, I ask if they sleep.
"They tell me that in time it will pass, but it doesn't pass."
Napoli move up into sixth place before their last 16, first leg tie back in their San Paolo Stadium on Tuesday.
The Serie A runners-up have won four of their last five league games, but are nine points adrift of a return to the Champions League next season.
Gattuso praised his team who just a month ago had slumped to 13th position.
"Often I was afraid, the team seemed to have lost faith," he said.
"I have been in the world of football for so long, it's never easy to switch the light back on. I saw the spirit I want, it was a great comeback.
"When I hear people talk about games like this as if they were a walkover, I get really angry. You have seen how difficult it is to play on these pitches.
"I have seen many videos of Brescia and we knew there would be a struggle. We won fighting up to the 94th minute and this is the mentality that I like.
"I saw the right spirit, the poison that we must always have."
Gattuso's side threatened early with Dries Mertens, who is just one goal off the club's scoring record of 121 goals, missing a chance after two minutes, hitting the woodwork.
But it was Brescia who broke through on their first opportunity with Chancellor getting his head to a Sandro Tonali corner to leave David Ospina without a chance in the Napoli goal.
It was the Venezuelan defender's third goal since joining from Qatari club Al Ahli this season.
Captain Insigne pulled Napoli level four minutes after the break from the spot after a Ales Mateju handball, with Ruiz curling in a stunning winner from the edge of the box five minutes later.
Mario Balotelli missed two chances to score, heading just wide before the break, and sending the ball over on front of goal after 72 minutes.
Brescia, second last in the table, extended their winless run to nine games going back to mid-December.
Leaders Juventus travel to SPAL on Saturday looking to extend their lead at the top of the table before next week's Champions League trip to Lyon.
Juventus are one point ahead of Lazio who play at Genoa on Sunday, with Inter Milan three points behind the champions before hosting Sampdoria.
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