It gave United a three-point lead over Liverpool heading into their meeting at Anfield on Sunday.
Despite continued speculation over his future, Pogba has played an increasingly important role in recent weeks to aid United's first serious title challenge since legendary former manager Alex Ferguson retired in 2013.
"I'm happy we are playing well and happy we are at the top, but there is a long way to go and we have big games coming up".
"We're ready, we're excited, we're hungry for it", Solskjaer said, "and it's a test of character and quality again". "He is a good character in the dressing room, he is always one the lads look to (because) he's a world champion. He's getting better and better every game".
The manner of victory against a Burnley side that had not lost at home since October also suggests United have the ability of champions to grind out a result when they need to.
The video assistant referee provided the main talking point of a tense opening period, with Burnley's last man, Robbie Brady, avoiding sanction for bringing down Edinson Cavani outside the area because of a foul by United left back Luke Shaw moments earlier at the other end.
The French World Cup victor dominated midfield in the second-half at Turf Moor and fittingly got the victor when his strike from the edge of the box took a deflection off Matt Lowton to wrong-foot Nick Pope.
"We couldn't be going there at a better time to play them", added Solskjaer.
The final whistle was met with roars from United players - especially from Pogba - as it was inside an empty Bramall Lane a few hours earlier after Sheffield United ended the longest winless start to a Premier League season with a 1-0 victory over Newcastle.
Michael Keane's 77th-minute header clinched victory, after Wolves midfielder Ruben Neves' volley in the 14th canceled out Alex Iwobi's first league goal for 17 months.
Everton won despite being without 14-goal top scorer Dominic Calvert-Lewin and fellow forward Richarlison because of injury.
"We adapted really well to the characteristics of the players we had", said Ancelotti.
The Blades still face an uphill task to haul themselves off the bottom of the table as they sit nine points adrift of safety.
"Sometimes [in the past] we looked like a team who wants to score as soon as possible, then we get panicked [if we don't score]". We're in a good position going into it'.