The two soldiers overcame a blunder on their second and final run, clipping a wall, to see off both American teams who took the other two medals.
Mark Grimette and Brian Martin claimed silver with Chris Thorpe and Clay Ives helping themselves to bronze.
The Germans had provided the foundations for their win by setting a track record in their first run, timing 42.953 seconds.
They came down slower in their second for an overall time of one minute 26.082 seconds, more than one tenth of a second clear of their nearest rivals.
"I'm so happy I could kiss everyone standing around me"
Alexander Resch, German luger
The Americans swapped medals from the last Olympics, when Thorpe, then partnered by Gordon Sheer, steered his luge to silver ahead of Grimette.
Thorpe had been in second place after the first run on Friday but his poor second run left the way open for the Germans to cruise to gold.
The fist pumping German pair, who have been in top form this season, winning the overall World Cup title, launched into wild celebrations as they crossed the finish line.
"I'm so happy, I could kiss everyone who is standing around me," said Resch, 22. "This is obviously the high point of our career."
His 24-year-old team-mate Leitner added: "We were under a lot of pressure to win gold.
"We had to win the national trials in Germany, eliminating a lot of other good teams to qualify, so to win is fantastic," he added.
For the US, Martin was delighted to win silver after his bronze in Nagano four years afo.
"It feels great to be silver. It feels good to take it higher than bronze," he said.