Republic of Ireland international Delap last scored a club goal against Fulham in March 2002 but this one was well worth the wait.
With a thunderous right-foot volley Delap put away a headed pass from Claus Lundekvam in the 64th minute to decide a match of many clear chances.
Tottenham played their full part in an entertaining and exciting game in which both goalkeepers made significant contributions.
For Southampton Antti Niemi kept out shots from Jermain Defoe and Jamie Redknapp and then from Robbie Keane after the former Leeds striker found himself clear with only the keeper to beat and ten yards out.
At the other end Kasey Keller was just as busy with important stops from James Beattie and Danny Higginbotham.
But Southampton should have done better with other chances, Beattie volleying over the bar while Kevin Phillips lifted a shot over Keller and wide of an open goal.
Fabrice Fernandes slid in unmarked at the far post but wasted a pass from Beattie as the Saints looked bright.
Tottenham had Gary Doherty and Christian Ziege booked for first-half fouls and were forced to rearrange their line-up at half time when Redknapp replaced Johnnie Jackson and Stephen Kelly came on for Ziege who limped off with a groin injury.
Redknapp was outstanding in midfield, relishing the room he found and giving a tough debut to 19-year-old Yoann Folly who had been drafted in as a defensive midfield player.
Ziege will have scans on the injury to determine its extent, leaving manager David Pleat to have no regrets about starting the match with three strikers in Defoe, Frederic Kanoute and Keane.
Southampton boss Paul Sturrock demanded an improved attitude and performance from his players after their poor display in defeat at neighbouring Portsmouth last week.
Saints responded with a much more dynamic performance but there is still, according to Sturrock, plenty of work to do before he can be satisfied that he has the team and the balance that he demands.
Man of the match - Antti Niemi: Southampton fans are used to his heroics every week and he made three saves to deny Tottenham goals, making them all look easy.