Left-back Robertson was only able to complete 62 minutes of the Premier League clash at Anfield, having to be replaced by Kostas Tsimikas after sustaining a muscular problem.
Meanwhile, Elliott - who came in from the start to replace the absent Fabinho - lasted 86 minutes before having to be replaced with a knock, and he was the victim of some crunching tackles on the night.
Speaking to the media after his side came from behind to record a narrow triumph, Klopp was confident that Robertson - whose injury was ironically triggered by a warm-up clash with Elliott - had only suffered a dead leg, and he does not believe that Elliott's issue will keep him out for too long either.
"Funnily enough, Harvey, before the game, injured Robbo a little bit. In our warm-up game, he hit him with a knee at the hip, so it was a dead leg," the Reds boss said.
"He could play but then at one point the muscle closed and he couldn't play on. Harvey got a knock, which was quite painful, but when you see the situation because it is clear that's painful, but that's it, I think."
Liverpool were punished for a slow start at Anfield within four minutes, as Kiernan Dewsbury-Hall strolled through the Reds backline to put the Foxes ahead, and Brendan Rodgers's side continued to cause Klopp's men problems on the counter.
However, Wout Faes then turned into his own net twice in the space of seven minutes to turn the game on its head, firstly deflecting a cross from Trent Alexander-Arnold over the head of Danny Ward before accidentally prodding home the rebound after Darwin Nunez had struck the post.
Faes became just the fourth player in Premier League history to score multiple own goals in one game, following in the footsteps of Michael Proctor, Jonathan Walters and ex-Liverpool man Jamie Carragher, and Klopp admitted that he had sympathy for the Belgian centre-back.
"Yeah, yeah, I had that once in my career - that a player scored two own goals. I really feel for him. Honestly, because it was absolutely unlucky," Klopp added.
"The first, how can it be more unlucky? And the second one, if you have ever played football, you go full speed in that situation, direction of the goal, you would expect the ball goes in.
"That's what your brain tells you and in the moment when you realise it is not going in, it is going to the post, you can't react anymore. These things happen.
"Yes, of course I feel sympathy - there is absolutely space for that, but whatever I say now will not change it. The boy will get through this, obviously, and Brendan [Rodgers] will help with that and everything will be fine."
Liverpool have now strung together a four-game winning streak in the Premier League, although they remain sixth in the table ahead of their opening match of 2023 away to Brentford on Monday.
The Reds have cut the gap to fourth-placed Tottenham Hotspur down to two points, but Manchester United - who have a game in hand on both clubs - could jump into the Champions League places if they beat Wolverhampton Wanderers on Saturday lunchtime.