First-half goals from Montel Gibson and Dominic Brown-Hill put Notts in control and John McAtee's 91st-minute strike wasn't enough for the visitors.
Notts fielded a strong side with eight first-team players starting to get more minutes under their belt.
Facing a young Shrewsbury outfit, it took less than five minutes for the Magpies to hit the front as Graham Burke's fierce strike was parried straight to Gibson, who fired in from close range.
It was 2-0 shortly afterwards, the ball breaking kindly for Brown-Hill 35 yards out and the youngster firing it spectacularly into the top corner.
Burke and Thierry Audel shot straight at the visiting goalkeeper, Stanley Aborah headed the ball over the top and Genaro Snijders blazed over from close range as Notts made the game attack against defence.
Wes Atkinson hit the bar for County on the hour before limping off with a knock, which left the Magpies to play out the remainder of the game with 10 men.
Shrews keeper Cameron Gregory palmed away a Burke free-kick and then superbly tipped a deflected shot from Gibson on to the crossbar.
Lewis Redding tested Adam Collin with a low effort and the visitors finally got a goal back in the final minute through McAtee, who saw his shot bobble into the bottom corner from 25 yards.
But there was not enough time left for the Salopians to rescue a draw, meaning Notts started their Central League Cup campaign with a victory.
Notts: Collin, Atkinson, Dunn, Audel (Jordon Richards 46'), Hollis (Dearle 55'), Brown-Hill, Snijders (Wildin 55'), Burke, Jordan Richards, Aborah (McMillan 46'), Gibson
Subs not used: Hall