San Jose got a second-round pick, 49th overall, and a fourth-round pick, 123rd overall. The Sharks also sent their fifth-round pick, 143rd overall, to the Devils.
The teams completed the deal just before the NHL's trade freeze for the Vegas expansion draft.
Mueller spent most of last season in the minors and has six points in 54 career NHL games. The Sharks risked losing the 22-year-old in the expansion draft for nothing. General manager Doug Wilson said this is an opportunity for Mueller to continue to develop.
The left-handed-shooting Swiss defender was the 18th pick in the 2013 draft. He tweeted that he was "excited to join" the Devils.
The Associated Press contributed to this report.