Germany midfielder Can found the net twice in the opening 21 minutes, either side of Mohamed Salah's first Anfield goal following his move from Roma, before substitute Mark Uth replied for the visitors.
"I do not have enough words. It is amazing," Liverpool manager Klopp told BT Sport. "It is 14 months of the hardest work and it feels amazing.
"A big, big, big compliment to my team. They played a perfect opening half an hour. I asked them at half-time to remember the feeling they had at 3-0 and to put it away."
Now unbeaten in 15 home games against German teams, Liverpool will learn their group-stage opponents in Thursday's draw in Monaco, which will feature five English clubs for the first time.
Liverpool join Chelsea, Tottenham Hotspur and Manchester City, who qualified via the Premier League, and Manchester United, who engineered their own Champions League return by winning last season's Europa League.
Liverpool will be making their first appearance in the Champions League proper since 2014-15, when they were eliminated in the group stage.
Klopp was once again without Barcelona target Philippe Coutinho, who is both ill and nursing a sore back, but the sizzling interplay of Firmino and Sadio Mane meant he was not missed.
The five-time European champions took a 10th-minute lead against continental first-timers Hoffenheim courtesy of the first of several delightful interchanges between their forward players.
Firmino's delicately weighted through-ball freed Mane and he back-heeled the ball into the path of Can, whose shot found the net via a heavy deflection off the sliding Havard Nordtveit.
- Qarabag breakthrough -
Egyptian winger Salah struck in the 18th minute, tapping in from close range after Georginio Wijnaldum's shot from Firmino's left-wing cut-back bounced back off the right-hand post.
Three minutes later another Mane back-heel yielded another goal, the Senegal winger's touch releasing Firmino, whose cross was adroitly volleyed home by Can.
Hoffenheim coach Julian Nagelsmann took swift action, sending on forward Uth for defender Nordtveit, and four minutes later Dejan Lovren's loose pass allowed the substitute to reduce the arrears with an angled drive.
After Wijnaldum and Mane had gone close, Liverpool added a fourth in the 63rd minute when skipper Jordan Henderson robbed his Hoffenheim counterpart Kevin Vogt and charitably teed up Firmino for a tap-in.
Wagner completed the scoring with a header from Andrej Kramaric's left-wing cross, but by then all hope of a Hoffenheim comeback had vanished.
"We didn't manage to implement our game-plan. We were running around like headless chickens," said Nagelsmann.
"We played well in the first leg, but today we didn't. Maybe the team was a bit too emotional. For the first 30 minutes, there were players who didn't seem to be on the pitch at all."
Sporting Lisbon thrashed Romanian outfit FCSB 5-1 in Bucharest, the visitors scorinh four second-half goals after Seydou Doumbia's 13th-minute opener was cancelled out by home defender Junior Morais.
Marcos Acuna, Gelson Martins, Bas Dost and Rodrigo Battaglia were all on target for Sporting, who blew their opponents away following last week's 0-0 draw in the Portuguese capital.
Dino Ndlovu grabbed the vital goal as Qarabag became the first team from Azerbaijan to reach the group stage, qualifying on away goals despite a 2-1 defeat away to FC Copenhagen.
Georgi Schennikov and Alan Dzagoev scored as CSKA Moscow downed Switzerland's Young Boys 2-0 in Russia to wrap up a 3-0 aggregate victory, while APOEL of Cyprus progressed 2-0 overall following goalless stalemate at Slavia Prague.