Williams scored on a 1-yard touchdown run and caught an 11-yard TD pass on a slant route from Tannehill as the Dolphins built an early 14-0 lead.
"When you score on two out of three drives, you feel like you're moving the ball well," said Tannehill, who has completed 82 percent of his passes through two games.
Panthers coach Ron Rivera said he was disappointed with his team's nine penalties, three of which helped prolong Miami drives.
"We had three big negative plays that were wiped out because we were offside and you can't do that and expect to be a good team," Rivera said.
Williams nearly had a third touchdown, but Dolphins coach Joe Philbin errantly challenged an official's call that the second-year running back was stopped before the goal line after diving for the pylon. When officials reviewed the play, they determined Williams actually fumbled the ball into the end zone.
Philbin said it was a "bad decision."
"The only person to blame is me," Philbin said. "I had the red flag, and I threw it. It was a bad decision."
Newton did move the offense to the Miami 2 on his final possession, but the Dolphins stopped them on four straight plays, including a scramble on fourth down by Newton in which he dove for the pylon but came up short.
"When you get inside the 10 you have to find a way to get it in the end zone," Newton said.
Copyright 2015 by The Associated Press