Real Madrid were made to toil by the 10 men of Athletic Bilbao but eventually won 3-1 on Tuesday thanks to two strikes from Karim Benzema to go level on points with La Liga leaders Real Sociedad.
Alvaro Negredo struck in stoppage-time as stuttering Valencia won 2-1 against his former club Sevilla on Sunday to earn coach Pako Ayesteran victory in his first home game in charge since succeeding Gary Neville.
Valverde's knowledge of Barca and his many years of experience of coaching in Spain saw him pip Argentine Jorge Sampaoli and Luis Enrique's assistant Juan Carlos Unzue to the top job.
Lionel Messi struck a spectacular free kick, scored another goal and set up two more in a 4-1 victory for Barcelona over nine-man city rivals Espanyol in the first leg of their King's Cup last-16 clash on Wednesday.
Holders Barcelona produced a second-half blitz to come from behind and beat Athletic Bilbao 3-1 to book a place in the King's Cup semi-finals after Atletico Madrid crashed out in a shock 3-2 defeat by Celta Vigo.
La Liga leaders Barcelona drew 0-0 at Huesca on Saturday and are nine points clear of second-placed Atletico Madrid, who cut the gap by beating Celta Vigo 2-0.
Real Madrid were dumped out of the Copa del Rey at the quarter-final stage on Thursday after losing 4-3 at home to Real Sociedad, while record 30-times winners Barcelona were sent crashing out of the Copa del Rey in a 1-0 defeat at Athletic Bilbao.
Lionel Messi was at the top of his game as Barcelona earned a 3-2 win at Athletic Bilbao on Wednesday to move up to third in La Liga.
News of all that's transpired on and off the football field
Liverpool frustrated by 10-man Arsenal in League Cup semi stalemate
Midfielder Dani Ceballos added: "It looks like us changing our coach worked. We have recovered some sensations we had lost and we have got the three points. Now our objective is to win all the games left and finish the season as well as possible."
A summary of Sunday's action in Serie A and Bundesliga.
IMAGES from La Liga matches played on Sunday
In other European football action Real Madrid were left frustrated after a goalless draw against Athletic Bilbao in their La Liga match on Sunday.
He made it 3-0 with a superb first-time finish into the bottom corner in the 20th minute, scoring his first ever treble in a competitive game, more than 19 years after making his professional debut.
La Liga leaders Barcelona handed Real Madrid more hope in the title race as they drew 0-0 at Athletic Bilbao on Sunday, only avoiding defeat thanks to some outstanding saves by Marc-Andre ter Stegen.
News of all that's transpired on and off the football field
Barcelona's Sergi Roberto and Robert Lewandowski struck just before halftime to give the league leaders a 2-0 home win over lowly Cadiz on Sunday in their pursuit of a 27th Spanish title.
Karim Benzema netted a second-half hat-trick as Real Madrid got back to winning ways with a 3-0 victory over Athletic Bilbao in La Liga on Sunday.
New Liverpool boss Juergen Klopp made it three wins from his last three matches with a 1-0 success at Rubin Kazan in Russia that left them second in Group B, two points behind Swiss club Sion on Thursday.
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News of all that's transpired on and off the football field
Images from the soccer matches played across Europe on Saturday.
An inexperienced Germany suffered an embarrassing 3-1 defeat to Slovakia in a storm-affected friendly ahead of Euro 2016.
