A young Southampton side overpowered their League One opponents with a first-half effort from Ramirez and a late header from Hooiveld.
The Uruguay international's early strike gave the Premier League team a perfect start, but their progress was not secured until Hooiveld's 83rd-minute goal.
Sean O’Driscoll made four changes to the City side that lost 3-2 to Swindon Town on Saturday, with Louis Carey (illness) and James O'Connor (thigh) among those left out, as Stephen McLaughlin was handed his full debut.
But Ramirez did get on the scoresheet soon after, as he opened the scoring on 15 minutes with a stunning volley from James Ward-Prowse’s flick-on.
Pochettino's side were well in charge of the game and could have doubled their advantage soon after when Lee was denied by City goalkeeper Elliot Parish.
The visitors then had a chance of their own as Jay Emmanuel-Thomas' shot was parried away by Kelvin Davis.
Guly then had the best chance of the early stages of the second half but he failed to guide his header from Ward-Prowse's corner past the alert Parish.
But Pochettino's side then confirmed their place in round four when Dutchman Hooiveld diverted Ward-Prowse's inswinging corner home.
