Celtic had a good opportunity to open the scoring after 20 minutes, but from Charlie Mulgrew's deep free kick, Van Dijk sent his free header wide of the far post from eight yards.
County's best chance of the half fell to Brian McLean, but the centre-back headed over from six yards out.
The miss proved costly, as Van Dijk gave the Scottish Premiership champions the lead on 42 minutes with a perfectly guided header from Emilio Izaguirre's deep cross from the left byline.
After the break, Scott Brown struck the post in the 52nd minute with a rasping effort, but from the resulting corner Van Dijk defender doubled his and his side's tally with another header.
Celtic restored their two-goal advantage just a couple of minutes later, though, when Ledley converted from Izaguirre's left-wing cross.
Ledely scored his second of the afternoon with a thumping header to wrap up victory 15 minutes from time and Nir Biton almost made it five, but Brown reacted smartly to tip the midfielder's header over the bar.