Liverpool's players applaud as Diogo Jota celebrates his goal by holding up a shirt with the name of team-mate Luis Diaz on it
Diaz missed the match following reports his parents had been kidnapped in Colombia, with officials still searching for the winger's missing father after his mother was rescued.
Diogo Jota, surrounded by his team-mates, held up the Colombia player's number seven shirt after firing the Reds into a 31st-minute lead when Matt Turner turned Darwin Nunez's attempt into the Portuguese forward's path.
"The best thing we could do is win the game for our brother," Klopp told Match of the Day afterwards.
"We played the game in the most difficult circumstances I've ever had. After more than 1,000 games [as a manager] you would think you've experienced everything.
"But it's not about us. We all pray that everything will be fine. The only thing we could do is fight for our brother and that's what they (the players) did. The boys fought for their brother."
Jota said the decision to hold up Diaz's shirt after scoring was "to show we're with him" in an "unimaginable situation".
"Luis was with us in the hotel [before the game] then he went home," he added to MOTD.
"He was going to play. I played instead of him and I showed him his shirt to show we're with him and we hope everything works out."
Liverpool were dominant and, after Jota's opener, Uruguay striker Darwin Nunez doubled the advantage four minutes later with his sixth club goal of the campaign after a cutback by the impressive Dominik Szoboszlai.
Forest's miserable day was complete when Turner and substitute Harry Toffolo got in a tangle, allowing Mohamed Salah to become only the third player in Liverpool's history to score in each of the opening five home league matches of a season.
Klopp's side are fourth is the table, three points behind leaders Tottenham, while Forest, who failed to seriously test Liverpool keeper Alisson, are 16th, five points above the relegation zone.
-- Courtesy of BBC Sport