The victory on Tuesday night takes Atalanta to within six points of pacesetters Juventus who continue their bid for a ninth consecutive title at Sassuolo on Wednesday night.
Gasperini's side have scored 93 league goals so far this season - 26 more than the leaders.
"We don't do it to humiliate the opponent but we try to play our game," said Gasperini.
"Already last year we were the best attack in the league, this year we have surpassed ourselves, scoring many goals with many different players.
"Playing in the Champions League has helped us and made us grow," added Gasperini.
Salvo
Pasalic opened the scoring within two minutes but Brescia responded quickly. Ernesto Torregrossa pounced on Mattia Caldara's poor clearance to level.
The hosts blazed back and were 4-1 up after half an hour with Marten De Roon, Malinovskyi and Zapata all netting in a five-minute blitz.
The bombardment continued in the second-half with Malinovskyi setting up Passaic for his second goal after 55 minutes.
The Croatian added his third of the night and his side’s sixth three minutes later sending in Ebrima Colley's cross for his ninth goal this season.
Highlight
"It's the best moment of my career," said Pasalic after his first senior hat-trick. "There are still five games left and we can do even better.
"We have scored more than 90 goals in Serie A. We have shown with the 2-2 draw against Juventus last week that we can match even the strongest teams."
Nikolas Spalek added a tad of respectability to the scoreline for the visitors seven minutes from time. But Brescia seem doomed to the second tier.
They occupy 19th place - the second of the three relegation slots - with 21 points from 33 of their 38 games and remain eight points behind 18th placed Lecce who take on Fiorentina on Thursday night.
Atalanta, by contrast, are having one of the best seasons in the club's 112-year history.
They are just a win away from qualifying for next season’s Uefa Champions League and they are in the last eight of this year’s reconfigured competition.
On 12 August they take on the French champions Paris Saint-Germain in Lisbon.
"We are preparing for PSG but also want to fight to the end in Serie A,” said Gaspirini.
"I think the best way to be ready for August 12 will be to play these last five league games to the maximum."
