Brazil international Alisson appeared to hurt himself taking a goal-kick in the first half at Anfield, with debutant Adrian coming on to replace him in the 39th minute.
Liverpool was 3-0 up at the time after a Mohamed Salah strike and Virgil van Dijk header added to Grant Hanley's own goal.
Divock Origi made it four before the break but Adrian was unable to keep a clean sheet, with Teemu Pukki getting the plucky Canaries on the board in the 64th minute.
Klopp, who confirmed in his news conference Alisson will miss the UEFA Super Cup against Chelsea on Thursday (AEST), felt Liverpool warranted the win despite some rust in its display and acknowledged the goalkeeping situation put a dampener on the result.
"In the first 60 minutes we were really good," Klopp said. "We could've controlled the game, that's clear. But it's early stages for us so you have to learn some things.
"I told the boys after the game everything is really good. If someone had told us before the game we'd win 4-1 we would've taken it, no doubt. But we should've controlled the game more, keep the ball.
"It's normal when you are not physically at your highest level that you lose concentration and pass the wrong balls. [Norwich] were brave…what they did was really good so we had to work a lot.
"But we won and deserved the three points and that's most important. I'm very happy with the start but of course the Ali situation is a shadow.
"He thought something hit him in the calf, so that's never a good thing — he thought that, but that's not how it happened, it's what he felt. That means it's probably a bit more than a strain. We have to make a diagnosis, we don't know it exactly, but it doesn't look too good.
"We brought [Adrian] in for situations like that, he was calm, did the job and is good for us in possession and his calmness helps us a lot.
"But we have a little problem. Number three [keeper] Caoimhin Kelleher is coming back but not 100 per cent fit yet, and even younger ones have some issues as well, so we have to think again."
Norwich rattled the upright through Moritz Leitner before Pukki's goal and had 12 shots across the 90 minutes, with head coach Daniel Farke praising its spirit in a tough opener.
"In general we are disappointed we lost the game. We wanted to start with some points but of course unbelievably happy with the character of my team. We showed great character," Farke said.
"We created more chances and had more shots on the goal in the second half. My feeling was in the first half we were not that bad. I love this team and I love my players.
"We have to stick to our beliefs. Of course we have to defend a bit more solid but my injured players are coming back and this was a lesson for some of my players because small details can make a big difference."