The Seagulls went into the game having not scored in their previous three league and cup matches but they found their shooting boots against a poor Barnsley side.
Brighton were 2-0 up after only 14 minutes with goals from Ashley Barnes and former England defender Wayne Bridge, with Barnsley keeper David Gonzalez at fault on both occasions.
Gonzalez, who played briefly for the Seagulls last season, dropped a free-kick by Craig Noone for Barnes to tap in after three minutes and Bridge beat the Colombian to his left with a low drive from 25 yards.
Barnsley pulled a goal back with a 36th-minute penalty from Craig Davies after Andrew Crofts fouled Jacob Mellis, but Craig Mackail-Smith got the final touch from a Barnes shot a minute later.
Record signing Mackail-Smith struck again four minutes after the break with a close-range header and Barnes completed the rout nine minutes from time by drilling in a cross by substitute Kazenga LuaLua at the near post.
Poyet said: "I think we have a better team because of the new players we have signed.
"We are much more aggressive and more dangerous going forward but we need to be solid as well. The key to it is to produce clean sheets.
"Everything went our way and Barnsley were the ones to pay. But it's about the next 43 games.
"It was a key game, a tricky one, and if something goes against you in a game like this you can be in a bad situation."
Poyet had no doubt that the key moment in the game arrived when Mackail-Smith restored the Seagulls' two-goal advantage moments after Davies had scored from the spot.
Added the Uruguayan: "The third goal was very important and credit to our players. They went into the game convinced it was a matter of time.
"Unfortunately for Barnsley they paid the price, they were an unlucky team coming up against us because everything went in our favour."
Barnsley manager Keith Hill admitted the best course for his team after this drubbing was to take their medicine and move on.
He confessed: "We had 12 players out but I have no excuses whatsoever. It was inept and I include myself in that."
Hill was disappointed that his men gave Brighton the opportunity to win the game more comfortably than they should have.
He said: "The opposition were excellent and it could have been up to 10-1. There was not too much I could say to the players as it was disappointing they did not accept the challenge before the game.
"We were up against two very good wingers but Brighton also had Bruno and Bridge, so it was like we were facing four wingers.
"But we can't change it. Sometimes you have to stomach it and move on. Hopefully we will learn and grow stronger."
Hill was keen to point out that Barnsley's budget does not match Brighton's and his club have been fighting against the odds to avoid relegation for the last "six or seven years."
He added: "We can't afford to sign the likes of Bruno, Bridge or Crofts who have come to Brighton.
"We can't go out and buy like Brighton and Hull. We intend to fight against the odds and for the last seven years we have been doing that."
