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Real moved on to 60 points after 31 games thanks to the victory but remain third in the standings.
Madrid are stuck in third on 57 points, 13 behind La Liga leaders Barcelona and five behind city rivals Atletico Madrid.
The last-gasp win left Madrid third in the standings with 57 points after 29 games, while second-placed Atletico Madrid have 59 and runaway leaders Barcelona are on 69.
Militao has agreed a six-year contract and will become coach Zinedine Zidane's first signing after his return to the Santiago Bernabeu this week.
The Frenchman won an unprecedented three consecutive Champions League titles with the Spanish giants before resigning in May last year.
Former player and reserve team boss Santiago Solari has been put in caretaker charge.
Cristiano Ronaldo signed for Italy's Juventus from Real in July for 100 million euros and has helped his new club to an unbeaten start in Serie A this season.
When the rumours about Cristiano appeared, Real Madrid players had bets among ourselves, said Modric.
Bookmakers William Hill reckon the Portuguese coach an 8/11 bet, with Southampton's Mark Hughes and Cardiff City's Neil Warnock the second-favourites to part ways with their employers at 7/1.
It was another frustrating outing for United and Mourinho, who finally took some responsibility for his team's poor form after defiantly defending his tactics and team selection throughout a patchy start to the season.