Henderson helped Klopp's men to a 2-0 win over Tottenham in the Champions League final in Madrid on Saturday.
The England midfielder has been the subject of criticism during his time at Anfield, where he arrived in 2011, and Klopp was thrilled for the 28-year-old.
"I'm happy for the boys. You know what people said about a couple of players of this team," he told a news conference.
"Jordan Henderson is captain of the Champions League winner 2019 – that's satisfying actually.
"That Millie [James Milner] did it at the age of 33. They are all very important, but I can say again, without Millie's dressing-room talks before the game – with a non-native manager – I think it would not be possible. It's so important."
A Mohamed Salah penalty and Divock Origi strike helped Liverpool secure their first trophy since Klopp took charge in late 2015.
Klopp was full of praise for his players, who led the club to their sixth European Cup/Champions League crown.
"All the things they did during the weeks, how they lifted when we had little downs, it's just incredible," he said.
"Tonight is really emotional, that's my main feeling, it's overwhelming, all that stuff. It feels really good but I'm much calmer than I thought I would be when it finally happened.
"It was not important to me to touch the cup or whatever. I loved the pictures when the boys had it, I loved it when I saw a few faces in the stands. That gave me everything I need.
"Tomorrow, going to Liverpool and having something to celebrate, that's big and I'm really looking forward to that."
While Klopp was more reserved in his celebrations at some points, he did break into song, an improvised rendition of the 1990s classic Let's Talk About Sex by Salt-N-Pepa, at one point following the victory .