The England international scored his first goals since August 23rd, firing in after eight minutes before adding a second against Mauricio Pochettino's men just after the hour mark.
And that was enough to lift the Reds into the quarter-finals of the competition, although Vincent Janssen pulled one back for Spurs on 76 minutes to set up a close finish.
Sturridge's talent 'never in doubt', says Klopp
On Sturridge's performance, the German explained: "That's his talent, that's his strength. He's a finisher. He's a really good striker. There's no doubt about this. I was never in doubt about this, even when he didn't score. That's not in question."
Klopp insisted that they "don't have to discuss the quality" of the striker, as he insisted that Divock Origi didn't score but "played quite well" as did Danny Ings, who he said "played really well."
He praised the 27-year-old for scoring "wonderful goals" which he said were "really important" for the team, insisting that Sturridge and "other players could have scored" and that then it "would have been perfect", but that it was still a "really good" performance from his team.
'Younger players wasted their energy'
Sturridge partnered Origi as Klopp used two strikers for the first time this season, and the Belgian was excellent as he showcased the kind of quality that made him first-choice frontman towards the end of last term.
The 21-year-old was forced off through injury with 22 minutes remaining after pulling up with cramp, the Reds boss acknowledging that his issue was nothing serious.
He said that "unfortunately" it was cramp and that it came when Origi was "in a good moment", adding: "That's the problem when you don't have rhythm, which he obviously doesn't have. Things like this happen."
Klopp noted that it was "quite intense" for his side and that they "wasted energy" because they "felt so good in this moment" that they "ran and created" but then they tired themselves out when they still "have to score."
"Unfortunately we didn't, so that means it's only a waste of energy," he insisted. But Klopp suggested that it was also "a good sign" even though it "can cause you cramps."
Origi has 'big potential', insists Reds boss
The manager was also asked what he said to Origi on his way off the pitch, and declared that he had "quite positive" words to tell the impressive No.27.
He added that there is "no discussion" about "the quality of Sturridge, Origi or Ings" and called it "only situation" as he said that they "cannot play with four strikers."
However, he called it "all good" and said that Origi has "big potential" and is "a good player" - insisting Belgium are a "lucky country" to have "this big number of [good] strikers."