The first chance of the afternoon fell to Norwich's Redmond. The winger cut in from the left and beat two defenders, but he fired his shot wide.
Bradley Johnson was the next to threaten for the home side, but his effort was off target.
However, Southampton could have had a penalty minutes later when Adam Lallana's cross appeared to hit Johnson's hand, but Howard Webb waved away the appeals.
Lallana had a great chance near the end of the first half, but struck the post.
Pablo Osvaldo forced John Ruddy into a save early in the second half when he shot on the turn.
Johan Elmander, making his league debut for Norwich, sent a header wide after more good work from Redmond.
The young signing from Birmingham finally broke the deadlock with a drilled shot into the bottom corner.
He then went close again, but after beating Calum Chambers, Boruc was able to save his low shot.