With Pochettino’s men languishing in 11th in the Premier League, an early FA Cup exit would have further ramped up the pressure on the Argentine.
Pochettino insisted on Tuesday he would not lose his hair over the stress of the job and was rewarded for some bold calls with his team selection.
Thiago Silva was dropped to the bench just hours after his wife apologised for a social media outburst in which she called for “change” at the club.
Raheem Sterling was also left out on a night when a number of Chelsea’s expensive young guns stood up to be counted.
“I think (it) was one of the best or maybe the best performance of the season,” said Pochettino. “It arrived in a very good moment for us.
“We have a young team. It’s about being solid and consistent and it is the most difficult thing in a young team.”
It was one of the club’s academy graduates who got the ball rolling as Gallagher smashed home from close range for his first goal of the season.
Jackson has been guilty of missing some glaring chances early in his Chelsea career, but was deadly with a downward header from Malo Gusto’s cross to double his side’s lead.
Villa were unbeaten in 17 home Premier League games dating back nearly a year until Newcastle won at Villa Park a week ago.
But Unai Emery’s men have exited both domestic cup competitions limply at home this season.
The tie was over before the hour mark as Fernandez’s stunning free-kick made it 3-0.
Rumours in recent days have suggested the Argentine World Cup winner could be looking for an exit, just over a year since his then British transfer record move from Benfica.
But Fernandez gestured to suggest he is staying put after curling past international teammate Emi Martinez into the top corner.
Pochettino has the chance to mark a difficult first season in charge with his first trophy in English football when Chelsea face Liverpool in the League Cup final on February 25.
And they will be confident of reaching the quarter-finals of the FA Cup when they host second-tier Leeds in the fifth round.
Villa got a consolation goal in stoppage time when Moussa Diaby fired in off the post, but it was too little, too late for the home side.
Forest earned the right to host United in the last 16 but needed penalties to see of Bristol City of the Championship.
Divock Origi fired Forest into an early lead, but Jason Knight quickly levelled for the visitors.
City were then left to rue missing a host of chances to win the game before extra-time and paid the price from the spot as Sam Bell was the only man to fail to score in the shootout.
© Agence France-Presse
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