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.
Mauricio Pochettino, meanwhile, changed his entire team from the 1-0 Premier League win at Liverpool on Saturday, with both Guly do Prado and Tadanari Lee making their first appearances of the season.
Jack Cork, who is yet to feature in the league this season, was back in the side after playing in the 5-1 second-round League Cup win over Barnsley and the 24-year-old was immediately involved as he looked to slip in Ramirez only to see his attempted pass cut out.
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.
And they could have drawn level before half-time when Derrick Williams headed over a free-kick from Brendan Maloney.
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.
Southampton struggled for rhythm in the second period though and the home crowd became increasingly restless as City pressed for an equaliser, with a good cross from Emmanuel-Thomas just failing to find the boot of McLaughlin.
But Pochettino's side then confirmed their place in round four when Dutchman Hooiveld diverted Ward-Prowse's inswinging corner home.
