A poor first-leg defeat was quickly consigned to history as Jonny Howson turned into his own net, Enzo Fernandez stroked in a near post effort, Axel Disasi tapped in a Raheem Sterling cross, and Cole Palmer tucked home a finish after a stray Boro pass.
In the second half, Chelsea were able to lower the tempo but still added more goals through Palmer and Noni Madueke to give the supporters a much-needed lift.
Pochettino needed it, Chelsea needed it
Victory buys everyone in the Chelsea project, including Pochettino, more time.
Boro manager Michael Carrick exposed flaws in Pochettino and his players by winning 1-0 at the Riverside. They risked causing a great upset after the first leg, but the Premier League giants set the record straight, as required by their demanding fans.
If they hadn't, supporters could have turned on the manager, the players, the sporting directors and even Todd Boehly and Clearlake Capital's project.
Now, everyone inside Chelsea has bought themselves time, until at least the cup final on 25 February. It will also offer another route into Europe, as three Premier League wins in a row put the club back improves the picture in the table.
Academy product Leo Castledine made his debut as Pochettino was able to ring the changes ahead of an FA Cup match at home to Aston Villa on Friday.
Chilwell's man of the match return
Ben Chilwell was the wildcard that unlocked the door for Chelsea against Middlesbrough.
The England international made his first start in almost four months. He was described by Mauricio Pochettino before kick-off as "one of our captains" and "one of the best footballers in the country".
His first attacking action was a clever run in behind, which could have won a penalty, as Tom Glover clumsily clattered into him during a high collision.
Just minutes later, he drifted into midfield and picked out a cute through ball to get Raheem Sterling in behind, forcing Howson's own goal. Chilwell was, of course, the captain on his return and led with his personality on the pitch.
Broja misfires while under pressure
Howson's own goal came after a heavy touch by Armando Broja, who could have scored himself but looked lacking in confidence.
After that, few chances fell his way, and he looked disappointed to be taken off in the 65th minute.
The 22-year-old striker could have done with a statement performance after it became apparent Chelsea wanted to sell him, albeit at an unrealistic £50million asking price.
Those reports will only dent his already waning swagger as he tries to rebuild form after major knee surgery.