The 3-1 result took Roma within a point of fourth place.
Enrique declared: "This is my Roma. This season there have been too many defeats, which is bad for the fans, the club and us, but this squad has character and wants to stay close to the teams fighting for Europe.
"I am happy for my lads, as they deserved this victory tonight. We have also lost games with Francesco, so you cannot say it was just Totti who changed the team. Besides, Lecce beat Catania this evening, so they are clearly not that bad.
"It was wonderful to see the faith of the fans during the game and they behaved wonderfully. The enthusiasm was not there before the match, but they helped us to get the better of a great team like Udinese.
"It's true we've had some terrible results, but we must've done something right to stay up there fighting for the Champions League.
"It's too easy for a Coach to say 'my team' is the one that wins! I take ownership of the heavy defeats too, as those are what help you to grow. We have players who are growing and in future will bring Roma great success, or at least I hope they will. There are many young lads here and that has to be recognised."