Tottenham’s motivation in hiring José Mourinho to replace Mauricio Pochettino was clear. They wanted the serial winner to guide them to silverware which, despite Pochettino’s excellence, has consistently eluded them. The drought pre-dated the Argentinian. Since the 1984 Uefa Cup, the club have won only three major trophies.

Mourinho has viewed this season’s FA Cup through the prism of opportunity. If he can win it, maybe it will galvanise his tenure. When Danny Ings put Southampton 2-1 up in the 72nd minute of this fourth-round replay, the prospect of Spurs advancing to a home last-16 tie against Norwich was outlandish.

Southampton had been the better team, particularly during a one-sided first-half. The cohesive football had belonged to them and so did the majority of the chances. Ralph Hasenhüttl’s tactics looked spot on and his players were stronger and quicker. They pressed with greater intensity. They seemed to be having fun.

Yet Spurs found a way to dig deeper. They did not play particularly pretty football but in the end it was winning football and to Mourinho that was all that mattered. Lucas Moura swapped passes with Dele Alli to score with a fierce low shot, having skipped past a substitute, Jannik Vestergaard and, as extra time loomed, Alli – who came off the bench to make a difference – curled a wonderful pass through for Son Heung-min.

He ran round the goalkeeper, Angus Gunn, who had hesitated before leaving his line, and everybody knew what was going to happen next. Gunn’s challenge was rash, Son went down and, in real time, it added up to a penalty for the referee, David Coote.

The replays showed contact was minimal, restricted to Gunn catching Son with a hand. Was it sufficient to send him tumbling? Son did not care about the debate. He picked himself up to score the penalty and Southampton found their best efforts were undermined at the last. Mourinho said that the best team had lost but Hasenhüttl did not want compliments, he wanted progress.

“It is not enough to dominate,” the Southampton manager said. “In the last 15 minutes we gave it away too easy. We didn’t defend well. I am not happy. It’s not about being the better team, it’s about being the winning team.”

Mourinho, who complained that injuries had restricted his options, sprang the tactical surprise, starting with a back three and partnering Lucas with Son up front. It did not work and he switched to 4-2-3-1 early in the second half. Only when Alli began to exert an influence towards the end did Spurs come into the game.

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Tanguy Ndombele had helped to put Spurs in front, shooting for goal after Jack Stephens had stopped Ryan Sessegnon with a last-ditch challenge and watching the ball deflect heavily off the unfortunate Southampton defender to beat Gunn.

It was tough on the visiting side, who might have taken the lead on two occasions. Ings took an excellent first touch to surge inside Jan Vertonghen but his shot was straight at Hugo Lloris – either side and he would have scored – while the Spurs goalkeeper appeared to misread a low shot from Nathan Redmond before saving with his trailing leg.

Danny Ings and Nathan Redmond celebrate after Ings put Southampton ahead in the second half. Photograph: Matt Watson/Southampton FC via Getty Images

Southampton advertised the equaliser. Ings rattled the crossbar after getting in behind the creaking Vertonghen and Lloris parried weakly from Ryan Bertrand’s long-range effort. The goalkeeper got away with that one. He would not be so lucky in the 34th minute. Sofiane Boufal found space and fed Redmond, who spun and shot for the far post. When Lloris pushed it out, Shane Long was all alone to sweep home.

Spurs were poor on the defensive cover and there were other instances of them giving up too much space, particularly in the areas between the wide centre-backs and the wing-backs. They did not move the ball with any conviction until the closing stages and their final decisions were poor. Southampton lost James Ward-Prowse to a grisly-looking knee injury but it did not affect their rhythm. Happily, the post-match update was positive, revealing only a deep cut.

Boufal miscued when gloriously placed after a burst from Ings on 50 minutes and Mourinho had seen enough. He swapped Vertonghen for Gedson Fernandes and changed his formation and, shortly afterwards, he sent on Alli for Ndombele. Vertonghen looked devastated and he brooded darkly on the bench, while Ndombele faded badly after a driving and trick-filled start.

Although Son broke and crossed for Lucas who headed wide – a big chance wasted – it was Southampton who hogged possession and controlled the tempo. It was difficult to isolate any star performer for them; it was a team display in the truest sense.

Vestergaard extended Lloris with a header from a corner and it was no surprise when Southampton took the lead. What a goal it was, Long’s clearing header from inside his own box releasing Redmond and the forward tearing into Spurs territory, skating past three opponents, before findings Ings, who stepped inside Japhet Tanganga and finished. Game over? It looked that way. Somehow Mourinho found a way.