After humiliating Premier League defeats to Wolves and Liverpool, Chelsea fans were fearing the worst having failed to break down the Villans at Stamford Bridge on January 26.
Pochettino also opted to drop star centre-back Thiago Silva amid a social media post from his wife appearing to call for the Argentine to be sacked during their defeat to Wolves.
In the lead-up to the replay, Pochettino insists clear-the-air talks with the player were held and his decision to put the defender who's played in all-but one of their 23 Premier League matches this season was purely football-based.
On his decision to bench Silva, Pochettino said pre-match: "I always pick my players based on performance and balance. That's it."
Silva was brought on in the 81st minute but the tie was won long before he entered the fray with Chelsea already 3-0 up by then.
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Conor Gallagher opened the scoring on 11 minutes with a thumping finish into the top corner, striking Noni Madueke's intelligent pull-back first time.
It was 2-0 ten minutes later when Nicolas Jackson, making his first start since returning from Africa Cup of Nations duty, caressed a header past Emi Martinez as he latched onto Malo Gusto's pinpoint cross from the right.
The clash was done and dusted with the goal of the night through Enzo Fernandez's sublime free-kick into the top left-hand corner of the goal from 25 yards out. Moussa Diaby pulled back a late consolation goal in stoppage time.
What was perhaps the most impressive part of the victory was Chelsea's ability to nullify the threat of Villa, who are very much in the conversation to finish in the Champions League spots.
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Ollie Watkins, who has 16 goals in 32 appearances this season, barely got a sniff as Axel Disasi and Benoit Badiashile showed there's potentially a bright future for the Blues in defence.
And maybe a brighter future for Pochettino, who's days in the job looked like they were about to be numbered after the loss to Wolves last Sunday.
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And in case you were wondering, Silva's wife, Belle, didn't say anything on social media while the match was going on.