Solskjaer has been put in charge until the end of the season after Mourinho was sacked on Tuesday and could not have asked for a better start in the Welsh capital thanks to Marcus Rashford's third-minute free-kick.
The visitors also had a slice of fortune when Ander Herrera's strike deflected in to double their advantage.
But United shrugged off Victor Camarasa's penalty, awarded for a dubious handball call on Rashford, with the sort of free-flowing play that appeared so scarcely under Mourinho as Anthony Martial started and finished a fine move also including Pogba and Jesse Lingard.
And Lingard struck twice more in the second half to secure United's biggest win of the season so far.
As well as getting Solskjaer off to a flying start, victory also sees United crucially close the gap on the Premier League top four to eight points thanks to Chelsea's 1-0 home defeat by Leicester earlier in the day.
Pogba was forced to watch the full 90 minutes from the bench last weekend as Liverpool inflicted the final blow to Mourinho's time in charge with a 3-1 win that left United 19 points off the top after just 17 games.
The French World Cup-winner was restored from the start for the first time in four league games and looked far more like the player United splashed a then world-record £89 million ($113 million) on in 2016 than the one that clashed with his former boss off the field and disappointed on it for much of the past two seasons.
Rashford gives United perfect start
It was Pogba who was felled on the edge of the Cardiff area early on and he stepped over the ball for Rashford to take aim with a dipping effort that deceived home goalkeeper Neil Etheridge and nestled in his bottom left-hand corner.
Solskjaer won only three of his 18 Premier League games in charge of Cardiff during an ill-fated spell that ended with relegation in 2014.
But a winning return to Wales rarely looked in doubt thereafter as the visiting fans enthusiastically chanted the Norwegian's name throughout.
The travelling hordes were enjoying their side moving the ball around with a speed and accuracy more reminiscent of Solskjaer's days as a player at Old Trafford when United dominated English football.
Pogba was at the heart of it and his intelligent pass picked out Herrera, whose shot flicked off Greg Cunningham to loop over the helpless Etheridge.
If Solskjaer's side were fortunate then, it was Cardiff who got a break nine minutes later when the assistant referee adjudged Rashford to have handled rather than control the ball with his shoulder inside the area.
Camarasa's perfectly-taken penalty could have sparked a collapse for a team short on confidence in recent times.
Instead, United responded resoundingly with a fantastic team goal as a quick interchange of passes between Pogba and Lingard freed Martial to score his ninth goal of the season.
Lingard took responsibility from the spot just before the hour mark after another soft penalty was awarded for a foul by Sol Bamba on the England international.
And the England international was the beneficiary of another defence-splitting pass from Pogba a minute from time as he rounded Etheridge and slotted into an empty net.