The young Swans fell behind against the Robins in the first half before a Ryan Bevan equaliser ensured the scores were level at the break.
A double from Jake Thomas in the second half sealed the win for the academy before the visitors pulled a goal back in the closing stages of the game.
Wednesday's result ensured the under-15s topped Group C South, which secures a home tie in the next round.
Under-15s coach Anthony Wright was pleased to progress to the next stage of the competition, but insisted there was room for improvement ahead of their next match.
"The main objective at this stage was to get through the group, so we are very pleased to progress to the next round and to top the group," he said.
"Despite our win over City, as a group we feel there are areas of our game we can improve on. We expect more from ourselves and will aim to deliver that when we play our next game in this competition."
The under-15s will now face a quarter-final against other teams that progress from their southern groups and could face the likes of Chelsea, Arsenal or Millwall.
U15 Premier League Floodlit Cup Results
vs Bristol Rovers (H) 5-3 Win
vs Reading (A) 1-5 Loss
vs Bristol City (A) 3-1 Win
vs Bristol Rovers (A) - 3-1 Win
vs Reading (H) - 0-1 Loss
vs Bristol City (H) - 3-2 Win