Hamilton came home ahead of Rosberg at the inaugural Russian Formula One Grand Prix on Sunday to stretch his lead to 17 points in the drivers' standings, while the result also clinched Mercedes' first team title.
And non-executive chairman Lauda expressed his delight at having halted Red Bull's run of four constructors' crowns in a row.
"It means a lot," he told the BBC immediately after the race. "I have to tell you I'm a little emotional.
"When we started this together - Ross Brawn, Toto Wolff and myself - it was hard work. It's an emotional moment, but let's think of the next race already.
"We will celebrate, but we have to go home. Lewis is flying home with me. Nico did a very good job after his first-lap mistake.
"I'm happy, I can't say any more than that."
Hamilton led from the second lap in Sochi after Rosberg locked up in overtaking his team-mate at the second corner and was forced to pit.
The German worked his way back through the field to finish second, but Hamilton was always in control as he claimed his fourth win on the bounce.
With three races left, and the last in Abu Dhabi offering double points, Lauda insists the rivalry between Hamilton and Rosberg will continue until the end of the season.
"They raced today, you saw at the first corner," Lauda added. "I thought that's it again, but it was not as there was enough run-off.
"They always race, they will continue to do this. Lewis is on a very good track, four in a row.
"It looks very good for him, it keeps the momentum."