The 28-year-old scored one and set up another in Les Bleus' four matches at the tournament, while also creating eight chances for his team-mates.
He completed nine dribbles – a tally bettered by only eight others, some of whom have played a game more – while his 51 passes into the final third is the sixth-most of any player.
Rooney believes a big part of Pogba's impressive performances was down to the movement of striker Mbappe, who ended the competition goalless despite having 14 shots.
And United's all-time leading goalscorer has challenged England forward Rashford to replicate Mbappe's positioning in order to unlock Pogba's best form at club level.
"Give Paul time and space and he has the ability and imagination to really hurt the opposition," Rooney said in his Times column.
"Mbappe had a poor Euros overall but he did contribute to Paul's success. Teams sat off France because they were worried about Mbappe's pace.
"That helped buy Paul that bit of room to play in, and Mbappe's ability to run in behind allowed Paul to play a type of pass that he is so good at, the brave throughball right down the middle of the pitch.
"At United, if Marcus Rashford can develop his game and be encouraged to start making those Mbappe-style runs in behind, I believe you would see Paul doing more of what he does for his country in a club shirt."
Rashford has featured from the substitutes' bench in four of semi-finalists England's five Euro 2020 games so far, but he has yet to score, assist or even have a shot.
At club level, Rashford scored 21 goals for United in all competitions last season and became the first English player to score eight times for the club in European competition in a single season since Bobby Charlton in 1964-65.