Maximilian Eggestein gave Freiburg an early lead at Berlin's Olympic Stadium before Christopher Nkunku equalised inside 90 minutes after Leipzig had Marcel Halstenberg sent off.
Leipzig's triumph comes having lost on their previous German Cup finals appearances to Bayern Munich in 2019, then Dortmund last season.
It is the first major title won by RB Leipzig, who are backed by energy drinks giants Red Bull and were only founded in 2009.
"That's crazy, we played 60 minutes with ten men," Leipzig managing director Oliver Mintzlaff told 'ARD'.
"We're going to party hard, this is an historic evening for us."
They shot up Germany's football pyramid with four promotions in seven years to reach the Bundesliga in 2016/17, twice finishing second to Bayern Munich in the last six years.
Leipzig coach Domenico Tedesco finishes his first season with a trophy having replaced Jesse Marsch last December.