The away side had the first chance of note, but Joe Healy sent his long-range thunderbolt wide 12 minutes in before Andy Cook found the side netting from Liam Ridehalgh's cross at the other end.
Rovers were awarded a penalty three minutes after the break when Connor Jennings was brought down in the box but James Norwood failed to convert as former Brighton goalkeeper Mitch Walker denied him an opener.
Tranmere's pressure finally paid off as they broke the deadlock in the 68th minute when substitute Stockton found space at the far post and got his head on the end of Ridehalgh's free-kick to hand his side their third straight win.
Match report supplied by the Press Association.
Match ends, Tranmere Rovers 1, Dover Athletic 0.
Second Half ends, Tranmere Rovers 1, Dover Athletic 0.
Andy Mangan (Tranmere Rovers) is shown the yellow card.
Substitution, Tranmere Rovers. Adam Mekki replaces Connor Jennings.
Lee Vaughan (Tranmere Rovers) is shown the yellow card.
Liam Ridehalgh (Tranmere Rovers) is shown the yellow card.
Connor Jennings (Tranmere Rovers) is shown the yellow card.
Substitution, Dover Athletic. Ross Lafayette replaces Moses Emmanuel.
Goal!  Tranmere Rovers 1, Dover Athletic 0. Cole Stockton (Tranmere Rovers).
Aswad Thomas (Dover Athletic) is shown the yellow card.
Substitution, Dover Athletic. Sam Magri replaces James Hurst.
Substitution, Dover Athletic. Aswad Thomas replaces Loui Fazackerley.
Substitution, Tranmere Rovers. Andy Mangan replaces James Norwood.
Substitution, Tranmere Rovers. Cole Stockton replaces Andy Cook.
Second Half begins Tranmere Rovers 0, Dover Athletic 0.
First Half ends, Tranmere Rovers 0, Dover Athletic 0.
Andy Cook (Tranmere Rovers) is shown the yellow card.
James Hurst (Dover Athletic) is shown the yellow card.
Joe Healy (Dover Athletic) is shown the yellow card.
First Half begins.
Lineups are announced and players are warming up.
Cole Stockton's second-half header proved enough as Tranmere beat Dover at Prenton Park to move level on points with National League leaders Lincoln.