Elanga climbed off the bench to rescue a draw for Man Utd in their UEFA Champions League round of 16 first leg against Atletico Madrid at the Wanda Metropolitano stadium on Thursday.
The 19-year-old also scored the insurance goal in a 4-2 victory over Leeds United in the Premier League over the weekend.
Elanga has seen his playing minutes improve ever since Rangnick took over from Ole Gunnar Solskjaer late last year.
The Sweden u-21 international was set to leave on loan in the winter transfer window but Rangnick is glad to have convinced him to stay.
"Of course, I had heard about his name before [I joined] but I didn't know him as a player because he didn't play for Manchester United [the first team]," Rangnick said.
"On the first training session, the night before the Crystal Palace game, he was the player with the best training performance, so I asked my colleagues and my staff members about him and then found out he had almost already decided to leave the club on loan.
"He came on against Palace as we all know and the week after I said, 'Listen, I want you stay and see how you do, how you are doing in training and the games until the closure of the transfer window'.
"I think right now he has just shown what is possible in football with the right mentality, the right approach to go into games. This is exactly what he did and what he always does in every training session."
Elanga has played 14 times and scored three goals for United this season.
