Klopp’s side had failed to score more than one goal in all of their last five matches – since a 3-0 victory over Southampton in mid September.
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But Roberto Firmino and Mohamed Salah were on target in the first-half, with the latter adding a third in the second half and Sadio Mane capping an excellent night.
Klopp has come in for some criticism for his conservative approach in recent weeks.
And while the German conceded his side started slowly, he said he always planned to go on the attack in the game.
“The plan was to start direct, we didn’t know how they would defend,” he said
“We wanted to be lively, but perhaps it [the indifferent start] went on too long. When the game calmed down we played football and created chances – it was a really good game and I enjoyed it a lot.”
Visitors Red Star played into their hands early on as they pressed in an attempt to recover from their 6-1 hammering at the hands of Paris Saint-Germain in their last outing.
Roberto Firmino was the early benefactor as he latched onto an Andrew Robertson cross before firing home after 20 minutes.
It was the Brazilian’s first goal in eight games. But he has now been involved in 19 Champions League goals since the start of last season with 12 strikes himself and another seven assists – more than any player in that time.
And Klopp’s side were in compete control on the stroke of half-time when Salah latched on to Xherdan Shaqiri’s through ball and blasted a right-footed shot into the roof of the net.
Shaqiri and Salah’s relationship seems to be blossoming – the Swiss star having also assisted his team-mate in Saturday’s 1-0 win at Huddersfield in the Premier League.
Salah grabbed his sixth goal of the season – and his 50th in all competitions for Liverpool – from the penalty spot after Mane was brought down in the 51st minute.
And the Reds winger said he was delighted to reach a half century for the club.
“It was an important game,” he said. “We knew the game was going to be tough. The most important thing was the three points.
“It’s great – I’m happy to score my 50th goal for Liverpool. We know how to play with each other and know our qualities.”
The Egyptian was substituted shortly afterwards and denied the chance of a hat-trick when the visitors conceded another penalty for handball in the 77th minute.
Mane stepped up in his place but fluffed his lines by putting his spot kick too close to Red Star keeper Milan Borjan, who turned his effort on to the crossbar.
But the Senegal star – returning from injury after breaking his thumb while on international duty – made amends by sliding home with ten minutes left on the clock.
And Klopp was delighted to see all of his front three on the scoresheet, while also reserving special praise for some of his summer signings.
He added: “It was good [the front three scoring], but it was good to see Adam (Lallana) coming on, the counter-attacking, the passes and Daniel (Sturridge) playing. It was a good football game and a deserved win.
“Fabinho was good from the start – Shaqiri was involved in everything. He is a really nice footballer and that helped. I thought he did really well.”
Liverpool are now in pole position to qualify from Group C after leapfrogging Napoli – who drew 2-2 with Paris Saint-Germain at the Parc des Princes – in to top spot.
Klopp’s side will travel to Red Star in match-day four, before a trip to PSG and a home clash with Napoli, where they will be looking to avenge a 1-0 defeat in the reverse fixture.
“It will stay exciting,” Klopp added. “This group is a tough one for all of us. We have to be ready every day and hopefully we will be ready in two weeks.”
CHAMPIONS LEAGUE RESULTS
Club Brugge KV 1-1 Monaco
PSV 1-1 Tottenham
Barcelona 2-0 Inter Milan
Borussia Dortmund 4-0 Atletico Madrid
Galatasaray 0-0 Schalke
Liverpool 4-0 Red Star Belgrade
Lokomotiv Moscow 1-3 FC Porto
Pasris Saint-Germain 2-2 Napoli