An own goal from Jack Stephens put Spurs in the lead on 12 minutes, but Southampton hit back through Shane Long ten minutes before the break.
Southampton took the lead in the second half 18 minutes from time, but Lucas Moura levelled five minutes later before Son Heung-min won and converted a spot kick on 87 minutes, which proved to be the winning goal.
Mourinho told BT Sport: “I have to be honest and say I think the best team lost on the pitch but my team were the ones with more heart and went to their limit.
“So many difficulties to even build a team, the options were not options to change the game. My team to deserved to win – but the best team lost.”
KEY STORY: Spurs confirm Champions League squad
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Spurs put a lot of energy in at the weekend when they beat Manchester City 2-0 on Sunday, and on Wednesday night it was evident that Saints indeed had an extra day’s rest as the visitors were the fastest out of the blocks.
The Spurs boss continued: “From the intensity level and freshness, they were much stronger than us. We tried to organise the team with the players available. They controlled us well.”
Spurs were outplayed in large parts of the game, but their strong finish saw them over the line.
Mourinho believes the 54th minute was the turning point, when he substituted Jan Vertonghen and introduced Dele Alli as Spurs changed from a back three to a back four.
“Dele changed the game after we switched to a back four,” Mourinho said.
“We had a link with Dele Alli, it was like trying to build a puzzle with a few pieces short.”
“It’s hard to sub players during the game. We were not controlling the game in a back five so had to change and he was the one to come out, that’s football.
“Some players react in a negative way but Jan acted like the professional that he is.”
Spurs set up a fifth round clash with Premier League bottom club Norwich City, and should they get past the Canaries, Mourinho will be just one win away from another trip to Wembley and three wins away from FA Cup glory.
Mourinho added: “To arrive here we had to play four matches, really difficult. We also played Norwich at home a couple of weeks ago and it was difficult.
“Let’s see a bit of potential for that game and to recover from injuries. After such a game against Manchester City, it was very hard.
“Dele could suffer pain and could cope with that tonight. We gave everything we could so that’s why I’m so proud of my team.”