Price, who went 22 of 35 with one interception, hit Campanaro with a 31-yard TD pass in the first quarter for Wake's first score. Dembry had six receptions for 96 yards and one rush for a 10-yard TD.
Mike Glennon had a career-high 315 yards passing and three touchdowns, all coming in the second half as the Wolfpack attempted to come back down 34-13.
The Wolfpack also had first-half field goals of 45 and 37 yards from Niklas Sade.
Price, who had 289 yards and three touchdowns against Syracuse last week, picked apart N.C. State's secondary in the first half with 223 yards on 17 of 25 passing. That helped stake the Demon Deacons to a 20-6 halftime lead.
Price completed his first eight passes Saturday, including Campanaro's 31-yard score, and a 19-yarder to Dembry on fourth-and-8 set up Newman's 32-yard field goal with 2:04 left in the first quarter.
N.C. State broke through early in the second quarter with Sade's 45-yard field goal with 14:13 remaining. But Price's 31-yard completion to Chris Givens on third-and-10 set up Wake Forest's next score, a 40-yard option pass from Campanaro to a wide-open Dembry with 13:02 left for a 17-3 lead.
Givens finished with 111 yards on six catches.
The Wolfpack made their furthest drive into Demon Deacons territory midway through the second quarter, moving to the 19-yard line before it stalled, leading to Sade's 37-yard field goal with 7:41 left.
Wake Forest's defense then came up big on N.C. State's next possession, stopping them at the Deacons' 32 on fourth-and-short with 2:44 remaining.
Price hit 5 of 9 passes in moving Wake Forest to the Wolfpack 12-yard line before the drive stalled. Newman then ended the first half with a 37-yard field goal for a 20-6 Wake Forest advantage.
Harris tacked on a 15-yard touchdown run and Dembry added his 10-yard reverse for a score late in the third quarter as the Demon Deacons pulled ahead 34-13.
But Glennon, who threw for 156 yards and a touchdown last week against Liberty in his first career start, got N.C. State on track in the second half.
Glennon, who hit George Bryan on a 19-yard yard TD for the Wolfpack's first touchdown in the third quarter, found Tobias Palmer on a 65-yard bomb later in the period, and hit T.J. Graham on a 63-yarder on fourth-and-10 with 5:34 remaining to pull N.C. State within a touchdown.