In his post match press conference, as you say, he criticised the way the defence played with the ball in the first half and said he asked the wingers to tuck in more. This change was reflective of the game as a whole. By surrendering the wide areas and making the centre of the pitch more compact he gambled that City wouldn't be able to trouble us too much if they had to resort to crossing. Further, by plugging the gaps in the middle Jose figured we could stop City playing through us. This was correct, City only played through us a couple of times after the second half started. Plus, that mostly happened after De Bruyne was introduced. The second half set up was very similar to what we did during the Liverpool game in March.
In addition to making the team shape more compact, Jose made use of Lingard's tactical awareness by having him move centrally when Lukaku ran the inside right channel. This put the City defence under a lot of pressure and you can see in the build up to our goals, the centre backs are concerned with whether to track Lukaku or Lingard. Getting our players closer together as a whole meant, when City defenders chose to try and chase a runner, there was space in the channels to exploit. Don't underestimate the role Lukaku and Lingard played as distractions as Pogba made runs from deep in the build up to our first two goals.
Also, according to Chris Smalling, Jose told them not to be the clowns at City's party. As Mou's on record telling Chelsea not to be the clowns at Liverpool's party four years ago, this sounds very believable. This, together with what other players are meant to have said e.g. Young and Carrick, inspired the team to just be better. This was just as important as the tactical switches. The players chose to not only believe in Jose's tactical adjustments but to also believe in their ability to execute those adjustments. You can have the smartest system in the world but if your players are giving away the ball under light pressure it won't matter.
The game showed the best of Jose Mourinho. He analysed where the plan was falling down in the first half and changed it. This is not the kind of plan that will work against everyone however. That's why its unwise to think doing the same thing will work every game. City play a style of football that demands they keep their wingers very, very wide. Pep also encourages his full backs to come into the inside channels and become auxiliary midfielders, which leaves space in behind. If you play compact against that you can get inside the channels. However, a lot of teams come to Old Trafford and play compact themselves. If we did the same thing against a lot of teams we'd just end up trying to play through them. Hence why Jose often resorts to 4-2-3-1 at home, he needs wingers to open up packed defences. That is also, I assume, why he is looking for new full backs who can give our attack more oomph.
Jose knows all this. That's why he will adjust his approach for every different match. That's why its good Jose manages United and not the GameVN.