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Goretzka's strike provided Germany with the late life-line.
The Italian replaces Zinedine Zidane, who left the La Liga club last month.
Atletico Madrid, Real Madrid and Barcelona are all very much in contention with only eight games each left.
Zinedine Zidane's side joined Atletico Madrid on 66 points and moved top of the table, although Atletico can reclaim first place when they visit Real Betis on Sunday.
Real moved above Barcelona into second place on 60 points from 28 games behind Atletico.
Real, who were knocked out in the last 16 by Manchester City last term and Ajax Amsterdam the season before, reached the last eight for the first time since 2018.
Danish forward Martin Braithwaite grabbed the decisive goal with a scrappy diving header early in the extra period.
Eden Hazard scored his first goal in almost three months while Karim Benzema bagged a brace as Real Madrid eased to a 4-1 win away to Alaves.
Lionel Messi and Antoine Griezmann each scored twice as Barcelona thrashed Granada 4-0 away to win a third consecutive La Liga game for the first time this season.
A towering header from Casemiro and an added-time strike from Karim Benzema saw Real Madrid beat a stubborn Granada side 2-0 at home.