The attacker scored both goals for Dan Leivers' team, who closed the gap on league leaders Mansfield Town to two points.
The game was rearranged from a month earlier when a serious injury to Rotherham's Revarnelle James forced an abandonment with Notts leading 2-0 after 37 minutes.
Leivers said: "It was potentially a tricky fixture for us due to the circumstances, but we were all more concerned for Revarnelle, who is a good friend to a number of our players. We are pleased to hear from Garreth Barker, Rotherham's academy manager, that Revarnelle is recovering well from the injury.
"We went into this game unchanged as, after a midweek friendly which we won 1-0 and all players took part to get some useful minutes under their belt, we felt that we didn't need to change anything in the starting line up from the 4-0 win at home to Bradford."
Following a goalless first 45 minutes, which the Notts head of academy coaching said was "a fair reflection of the half", the Millers took the lead as Joe Searson-Smithard's clearance fell kindly for a Rotherham attacker to slot into the net.
But the young Pies were level soon after with Campbell drawing a foul in the penalty area and dispatching the spot-kick himself into the bottom right corner.
Notts were on top and almost completed the turnaround on a couple of occasions as Peter Dearle headed narrowly wide of the upright and Dominic Brownhill fired just past the post.
Leivers felt his team should have had two penalties - Campbell was tripped as he was about to shoot before Harry Gibbons was chopped down inside the penalty area.
But County were not to be denied a winner and Campbell proved to be the hero of the day, coolly finishing past the Rotherham goalkeeper following a strong burst from midfield by the effervescent Alex Howes.
With the final whistle signalling three wins in a week for his side, Leivers was pleased with the squad's progress and mental fortitude, saying: "Although the victory could have been more comfortable, we managed the game really well in the final 10 minutes and Rotherham's only shot on goal was from distance and that went well wide of the post.
"The back four of Nathan Cobain, Jordon Richards, Peter Dearle and Declan Dunn were as solid as I've seen all season, which left Joe Searson-Smithard relatively untested. We were very well organised defensively and looked like we could cause problems on the counter attack.
"We were really pleased with the character of he lads, not only coming back from 1-0 down to win the game but also under the circumstances of the rearrangement."
Notts: Searson-Smithard, Cobain, Richards, Dearle, Dunn; Clayton-Naute, Brownhill, Howes, Osborne; Campbell, Gibbons (McCartney, 70').
Subs: Towers, Hall, Ebanks, Robinson.