Son late show sends Spurs into FA Cup fifth round
Jose Mourinho's side were 12 minutes away from being knocked out after squandering the first-half lead given to them by Jack Stephens' own goal.
Shane Long and Danny Ings scored as Southampton hit back to take the lead at the Tottenham Hotspur Stadium.
But Lucas Moura equalised before Son won and converted the decisive penalty with just two minutes left.
Tottenham, who last won the FA Cup in 1991, will host struggling Norwich in the fifth round in the first week of March.
Aiming for the first trophy of his reign, Mourinho, who replaced Mauricio Pochettino earlier this season, will be relieved Tottenham avoided a surprise exit from a competition he has not won since 2007 with Chelsea.
Tottenham are without a major trophy of any kind since the 2008 League Cup and Mourinho would dearly love to end that drought, but his team will have to improve significantly on this erratic display.
With Premier League teams enjoying a first winter break in the English top flight, this was a replay neither side wanted.
Every team should have 13 and 16 days between games, but Southampton's break has been cut from 14 days to 10, with Tottenham's cut to 11, as a result of the replay.
Mourinho had more reason than most to bemoan the additional fixture with Giovani Lo Celso and Erik Lamela ruled out and Dele Alli only fit enough for the bench after his injury in Sunday's win over Manchester City.
Sofiane Boufal's late equaliser saved Southampton in the initial meeting at St Mary's after Son had put Tottenham ahead.
Southampton were more impressive this time and almost took the lead when Long backheeled into the path of Nathan Redmond and the winger's low shot forced Hugo Lloris to save with his legs.
But Tottenham had won eight of their last nine home games against Southampton, including a 2-1 success in the Premier League earlier this season, and they were in front after 12 minutes.
- Momentum swing -
Ryan Sessegnon's burst opened up the Southampton defence and when the ball bounced to Tanguy Ndombele on the edge of the area, the midfielder drilled a powerful shot that took a big deflection off Stephens into his own net.
Ings should have equalised when he ran onto James Ward-Prowse's pin-point pass and got to the ball ahead of the hesitant Lloris, but the in-form striker's shot cannoned back off the bar.
Southampton's 34th-minute equaliser carried more than a whiff of blame for Lloris, who limply parried out Redmond's shot for Long to slot home.
There was a blow for the visitors before half-time when Ward-Prowse was stretchered off after needing oxygen for a knee injury suffered in a challenge with Sessegnon.
Hasenhuttl's side deservedly took the lead in the 72nd minute.
Redmond made an incisive surge from deep inside his own half and slipped a pass to Ings, who cut past Japhet Tanganga before curling a fine finish into the far corner from 18 yards for his 17th goal of the season.
But Lucas came to Tottenham's rescue in the 78th minute when he took Alli's pass and cleverly turned into the space on the edge of the area before drilling a superb strike into the far corner.
The momentum had swung in Tottenham's favour and Son won it in the 88th minute when he charged into the area and went down under minimal contact from goalkeeper Angus Gunn.
Son stepped up to take the penalty himself and fired home to send Tottenham into the last 16.