The five-year deal, which runs through the 2023 season, was approved by the school's board of trustees on Thursday morning and is subject to approval by the North Carolina Board of Governors.
Drinkwitz, 35, replaces former Mountaineers coach Scott Satterfield, who left to become Louisville's coach.
"I'm excited to welcome Eliah and his family to App Nation," athletic director Doug Gillin said in a statement. "The App State football program is in great position to continue its championship ways. Eliah is committed to excellence academically, competitively, socially and to the well-being of our student-athletes.
"... Eliah has great character and a clear vision for the future of Mountaineer football. I'm excited about the future of our football program."
Drinkwitz has guided NC State's offense the past three seasons after spending the previous two seasons as an assistant coach at Boise State. He also worked at Auburn and Arkansas State.
"My family and I are excited to embrace and build upon the proud tradition of Appalachian State football," Drinkwitz said in a statement. "... This is a special time for App Nation, and we will work tirelessly to uphold the championship tradition."
This past season, the Wolfpack ranked No. 1 in the ACC in passing (327.7 yards per game) and tied for third in scoring average (35.6 points) and 12th in rushing (143.7 yards). NC State quarterback Ryan Finley led the league in passing, completing 67.9 percent of his attempts for 3,789 yards with 24 touchdowns and 9 interceptions.
Drinkwitz coached Boise State quarterback Brett Rypien when he was a freshman in 2015, and Drinkwitz was a quality control assistant when Auburn won a national title in 2010.
Satterfield went 51-24 in six seasons with the Mountaineers, guiding them to a share of three Sun Belt Conference titles and three straight bowl victories.
Appalachian State plays Middle Tennessee in the R+L Carriers Bowl in New Orleans on Saturday (9 p.m. ET, ESPN/ESPN App).