The 24-year-old forward, who had been with Salzburg since January 2015, scored 16 goals and recorded 22 assists in 40 games in all competitions for the Austrian side this season.
"Hwang Hee-chan fits our profile perfectly," Leipzig sporting director Markus Kroesche said in a statement.
"He can play anywhere in attack, on the wings or through the middle. He gives us even more flexibility, with his pace and movement.
"Hwang Hee-chan had a brilliant season in Austria, impressing as a goalscorer and provider, and in the Champions League."
Hwang has played 32 times for South Korea, and was also part of the Olympic squad in 2016.
"I'm really looking forward to the challenge of playing for Leipzig and in the Bundesliga," Hwang said.
"I'm joining a young club with ambitious goals and a team hungry for success. They're a great fit for me, as is their attacking style of football."
Henrichs, 23, began his professional career with Bayer Leverkusen in 2015. He completed a switch to Monaco in 2018 and has made 44 appearances for the French club.
"After two interesting years in France, I'm delighted to be back in the Bundesliga," said Henrichs, who has won three caps for Germany.
"The club has ambitious goals and the team plays a clear style of football that I was already impressed by in my four games against them with Leverkusen."