As Madrid kept losing defenders through injury, Celta right-back Hugo Mallo brought his team back into the game, but Sergio Ramos' Panenka penalty and Dani Ceballos' 25-yard screamer sealed the three points before Brais Mendes' consolation strike in added time.
However, league rules state that an interim coach can do a job for only two weeks - a period that ends on Monday - meaning he has to be registered or drop back to his Castilla role.
"He has done excellently", Madrid club director Emilio Butragueno said on Sunday night. "I don't just say that now as coach of the first team".
"He's one of the greatest things we've ever had at this club", Solari said last month.
After the global break, Madrids generous run of games continues with league matches against Eibar, Valencia, Huesca, Rayo Vallecano and Villarreal before Christmas, none of whom now reside in La Ligas top half. "I usually don't plan too far ahead".
The upturn has left Madrid only four points adrift of league leaders Barcelona and on the brink of qualification for the last 16 in Europe.
"The players are the ones who go out onto the pitch", Solari said at the post-game news conference.
'Casemiro had to come off, Gareth made a huge effort because the kick he received was causing him problems.
Europa Press cite a club source as saying: "It's true that we initially stated that Santiago Solari would take over the coaching position in a provisional capacity after Julen's departure but the club is very happy with his performance and he will continue".
"There is no doubt that he came at a very particular moment and has been able to achieve very good results". "Above all today, in a very hard game".
"His future? We are happy".