In a frenetic, end-to-end clash at a sold-out Carrow Road, the hosts' 19-year-old winger proved the difference as he capped a fine performance with the winner.
Redmond's pace often had Southampton on the back foot and saw him score the decisive goal in the second half, cutting in from the left to unleash a fierce drive from the edge of the box.
The match could have ended quite differently had it not been for the earlier contentious call from referee Howard Webb.
Bradley Johnson clearly handled an Adam Lallana strike in front of Webb, but the referee waved away those claims.
More top news
Thursday's weather for the west of the region
Thursday's weather for the east of the region
Thursday morning's weather for the west of the region