Sturridge was a surprise inclusion in the starting line-up, his first start for Liverpool this season.
Klopp deemed Firmino’s eye injury too severe for him to keep his place, but was named on the substitute’s bench.
And the Liverpool boss’ decision paid off when Sturridge put Liverpool in front with a close-range header.
James Milner doubled Liverpool’s lead from the spot before Thomas Meunier pulled one back before half-time.
It looked as though Liverpool had thrown it away when Kylian Mbappe netted the equalised after Mohamed Salah sloppily gave the ball away in the middle of the park.
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However, Firmino came off the bench to rescue all three points for Liverpool to give Klopp the perfect start to their Champions League campaign this season.
For Sturridge, it was also a chance for him to prove his worth to the squad after being shipped out on loan last season.
It was the first Champions League start for Sturridge since 2012 during Chelsea’s European winning campaign.
But he remained grounded when talking about Liverpool’s chances of going one step better than last year when they lost to Real Madrid in the final.
“The last time was for Chelsea when we won it, so it is great to contribute to the team today, but it’s early days so we aren’t going to get ahead of ourselves,” Sturridge said.
“It is a team here and everybody is going to add to the wealth of the club.”
He added: “It was a great to score and it was a great ball by this man here [Robertson].
“Him and Trent (Alexander-Arnold) have got a competition going on, so it was good to help him out.”
Liverpool now sit top of their group after Red Star Belgrade and Napoli played out a 0-0 draw.
The Reds return to Champions League action in two weeks’ time when they travel to Italy to take on Napoli.
Before that they face Southampton and Chelsea in the Premier League, sandwiched with a Carabao Cup against the Blues a week tomorrow.