Cole, playing in his 600th career NHL game, has five goals and five assists in February.
Pascual Dupuis scored for the Penguins.
Cam Ward stopped 33 shots for Carolina, and Marc-Andre Fleury had 20 saves for Pittsburgh.
A couple of veterans were making their first appearances with their old clubs since Thursday trades, with Cory Stillman back in the lineup for Carolina and Alex Kovalev skating for Pittsburgh.
Pittsburgh played its 22nd straight game without captain Sidney Crosby (concussion), and the club gave no update on his status.
The Penguins won the four-game season series between the clubs with three wins, one in overtime.
Harrison put the Hurricanes ahead quickly, beating Fleury from the high slot on Cole's pass from the left point at 1:44.
Corvo doubled the lead on the power play at 16:43, taking a pass from Joni Pitkanen on the left point and one-timing a blast past Fleury.
Cole made it 3-0 at 10:59 of the second, receiving Staal's pass out of the right corner on the left doorstep and tipping it in.
Dupuis got Pittsburgh on the board at 15:32 of the second, picking up a rebound of a Michael Rupp shot and slamming it by Ward on the stick side.
But Jokinen got the clincher on a rush at 14:53 of the third, beating Fleury stick side from the slot with Brandon Sutter on the helper.
Notes: Hurricanes D Joe Corvo played in his 200th game with the club, while Hurricanes RW Troy Bodie played in his 100th NHL contest. ... Pittsburgh D Paul Martin missed his third straight game (upper body). ... The game was a sellout, Carolina's fifth in the building this season.