Stockton, who inspired Tranmere to victory over non-League Braintree in the first round, dragged Rovers back into the game before the break after Liam Cooper had given the Spireites the lead.
And then 12 minutes after the interval McGurk pounced on Bolton loanee Michael O'Halloran's throughball and coolly slotted home to keep Tranmere marching on in both the league and cup.
Tranmere captain James Wallace departed early through injury but Rovers responded positively as Stockton was twice denied by Chesterfield 'keeper Tommy Lee.
Danny Whitaker brought a fingertip save from Owain Fon Williams at the other end before Cooper opened the scoring just after the half hour from Luis Boa Morte's resulting corner.
Lee stopped O'Halloran from equalising moments later but he could do nothing with Stockton's effort from the edge of the area in the 42nd minute. O'Halloran then fed McGurk after the break before being thwarted by Lee again 10 minutes from time as Rovers looked to put the result beyond doubt.