Goals from Gary Cahill, John Terry and substitute Demba Ba in the second half secured the three points that lifted Chelsea up to second in the Premier League.
After a poor start to the game, the Blues really clicked into gear after half-time, with the introduction of Ba a real turning point in the game, as the former Newcastle man provided an assist and got a goal late on in the game.
The game provided some insight as to how the festive period might go for both sides, and here are six things we learned from the game.
Pochettino vs. Mourinho: a tactical battle of two halves
The young Argentine coach had never beaten Mourinho—they faced each other while at Espanyol and Real Madrid, respectively—but he used his tactical wit to outsmart Chelsea in the first half.
From the kick-off, the Saints pushed five men into the opposition half and moved the ball forward immediately. Of course, Michael Essien's backpass was disastrous in every sense, but the sheer work rate and attacking movement from Pochettino's side meant they were able to capitalise on the error, allowing Jay Rodriguez to score.
The 38-year-old certainly did not hold back with his formation, playing James Ward Prowse, Adam Lallana and Rodriguez behind Daniel Osvaldo, with the midfielders causing all sorts of problems in the first half.
The defence looked organised and did a sterling job in limiting Fernando Torres to just the one chance, although it was a fine save from Artur Boruc to deny the Spaniard.
The second half was a different matter altogether, with Chelsea throwing every attacking resource they could at Southampton in Frank Lampard and Demba Ba, with both having some involvement in John Terry's and Gary Cahill's second-half goals.
Pochettino may have started well, but Mourinho's response to going down was superb, and the tactical mastermind eventually pulled through to win the war.
Juan Mata must start
The little Spaniard was brought back from the abyss and into the starting lineup after Mourinho wielded the axe on John Obi Mikel, Willian and Frank Lampard, and unsurprisingly, it paid dividends.
The 25-year-old was given a floating role in the playmaking position as Oscar made his way out to the right, and together with Hazard, the trio really gave Southampton's defenders a lot to think about.
Mata, in particular, was not only looking to get the ball, but uncharacteristically fought back to regain possession also. And that's the effect of Mourinho—beforehand, this was one aspect of Mata's game that was missing.
Chelsea look like a different team when he is playing. The ball moves quicker, the players are in better positions and there is generally more creativity.
Mata's two assists on Sunday were vital in the 3-1 victory, and to leave him out after this would be a criminal mistake from Mourinho.
Petr Cech's lack of form a concern
The Czech goalkeeper was never really in doubt of losing his place in the Chelsea team, and one would think that under Mourinho, the chances of that changing were almost unheard of.
But his error against West Brom, combined with a shaky performance in general, showed that perhaps the 30-year-old is showing signs of decline in his consistency.
Essien's catastrophic backpass was poor and unnecessary, but Cech's lack of confidence in coming out to get the ball allowed Rodriguez to surge forward and slide past him.
Perhaps it's a coincidence, but with Thibaut Courtois coming back from his loan spell at Atletico Madrid next season in fine form, Cech could be looking over his shoulder at the 21-year-old.
Ba may yet have a future at Chelsea
Speculation has linked Ba with a move away from Chelsea in the January transfer window, per The Independent, but based on his superb cameo from the bench, Mourinho may think twice about letting the Senegal international leave.
The 28-year-old entered the action after coming on to replace Essien in the second half as Chelsea chased an equaliser, and his aerial presence caused problems for the Southampton defence immediately.
Fernando Torres linked up with the forward, feeding off his lay-offs as he confidently strolled toward goal, but Ba was the main threat.
From a corner, the ball came to Ba and he stretched to place his effort onto the post, with Gary Cahill heading home the rebound. It was his instinct to position himself at the far post which made the difference and gave Chelsea a way back into the game.
Southampton: Maintain intensity and they'll be fine
One aspect of pressing high up the pitch is that it commands a physical effort throughout the entire 90 minutes, and unfortunately for Southampton, they could only manage it for 45.
Morgan Schneiderlin and Victor Wanyama pulled the strings in the first half, closing down Ramires and giving Michael Essien a lot to think about in terms of his own career after being bossed in the first half.
But when Frank Lampard, a better passer than Essien, entered the field, it was a different game altogether. The Saints were having to work harder off the ball to close down, opening up space for the likes of Hazard and Juan Mata to drop into.
Eventually, fatigue overcame the players and they invited Chelsea onto them, with the Blues in full throttle. The lack of closing gave them time to choose their next pass and it ultimately cost Pochettino's side.
If they can keep up the high pressing and fast-paced football, there's no doubt it's a workable tactic for them to use.
Essien is no longer up to scratch
It's unfortunate that injuries can ruin a player's career, and Essien is no stranger to the cruelty that can come with a long spell on the sidelines.
Even in spite of his loan spell at Real Madrid last season, where he made 34 appearances and scored two goals, he showed his lack of sharpness and pace.
With John Obi Mikel, Ramires and Frank Lampard all in front of the 30-year-old, it seems difficult to look past a summer exit for Essien, whose contract expires this summer
A shame, given that Essien has been at the club for eight years after a £26 million move from Lyon, but the fact that he almost cost his side a chance to grab victory could be an indicator that perhaps he cannot be relied on.