Varney, recalled by the former Preston boss after his loan spell from Derby expired, cancelled out a Rory Fallon strike, with the Sheffield side making it four wins from five since the departure of Brian Laws.
Wednesday had to work hard for the points against a resolute Plymouth side, who were lifted by a 1-0 win over Derby in midweek.
They set the early pace with Jamie Mackie drilling in a long-range shot that flew narrowly wide before Fallon almost found a way through after stretching to meet Chris Clark's cross.
They snatched the lead in the 23rd minute when Wednesday failed to cut out an Alan Judge corner.
Fallon rose unmarked to head past Lee Grant from close range for his fourth of the season.
Wednesday were back on level terms a minute later after Tommy Spurr's long throw from the left was flicked on by Marcus Tudgay and Varney added the finishing touch from six yards.
Varney added a second on 42 minutes with Tudgay again the provider as the Wednesday striker out-sprinted Reda Johnson to cut inside and beat David Stockdale from the tightest of angles.
Spurr saved Wednesday on 63 minutes after sliding in to steer Jamie Mackie's low cross clear from under the nose of Fallon as Plymouth stepped up a gear.
Mackie could have levelled matters as he exposed a gap in the Sheffield rearguard but he hammered the ball high over the target.
Wednesday clung on in a nail-biting finish to claw themselves further away from the drop zone.