The goal gave Argentina all three points and condemned their neighbors to a losing start to their Copa campaign.
Barcelona earned a 1-1 draw with holders Sevilla in the first leg of the King's Cup last-16.
Images from the football matches played across Europe on Sunday
The clinical Argentine Messi now has 11 goals in his last five games in all competitions and with record signing Ousmane Dembele injured and Luis Suarez rested, he put on a sensational show at Camp Nou to extend Barcelona's La Liga lead.
Joining them in the last four were Getafe and Deportivo Coruna.
Real were smarting from a surprise 1-1 draw at struggling Elche in their last match and were missing captain Sergio Ramos, who was out with a stomach bug.
The English champions will go through to the last 16 of the Champions League if they overcome Real Betis in Seville.
David Beckham was sent off in the final minute of Real Madrid's 0-0 home draw with Real Betis
Real Madrid were dumped out of the King's Cup by Primera Liga strugglers Real Betis on the away goals rule after they were held to a 1-1 draw at the Bernabeu on Thursday.
Morata, who had scored his first goal for the club in last week's 2-0 win over Villarreal, rose to head home from close range on the half-hour mark following a corner which had been flicked on by Atleti captain Diego Godin.
Ronaldinho was in inspirational form as he returned to lead Barcelona to a 5-1 home victory over an unfortunate Real Betis in the Primera Liga on Saturday.
Midfielder Mahamadou Diarra scored his first goal for Real Madrid to earn a dour 1-0 win at Real Betis.
With United unable to sign a replacement, the pressure will be on 21-year-old England forward Marcus Rashford to deliver the goals in a more central role than he has previously operated in.
Barcelona eased to a 3-0 win over Racing Santander to keep pace with leaders Sevilla in the Primera Liga.
A Thierry Henry special and two trademark free kicks from Ronaldinho earned Barcelona a 3-0 win at home to Real Betis that put them within a point of leaders Real Madrid at the top of the Primera Liga on Sunday.
Both sides had already qualified for the knockout stage of the Champions League and all that was at stake was first or second place in the group.
AC Milan scored a 4-0 win over Fenerbahce with Andrei Shevchenko scoring all four goals in their Champions League Group E victory.
Liverpool turn to the relative calm of their Champions League defence with a place in the last 16 at stake on Tuesday.
Manchester United beat Hungarian champions Debrecen 3-0 in the first leg of their third qualifying round tie on Tuesday.
Barcelona's grip on top spot in the Primera Liga was strengthened when Real Madrid were held goalless by Real Betis and Valencia lost 2-1 to Racing Santander.
Brazilian defender Filipe Luis scored for the second game running as Atletico Madrid beat Real Sociedad 1-0 on Tuesday to rack up a fifth straight win in La Liga and provisionally consolidate third place in the standings.
The Brazilian playmaker returned from a two-game suspension at the weekend and scored twice in a 5-1 drubbing of Real Betis.
Barcelona surrendered lead to arch-rivals Real Madrid after they were held to a 1-1 draw at home to Real Betis.
"We have nothing to argue with, we have no defence because our performance in the first half was so bad," Mourinho said.
Yet Milan's defensive frailty was punished when Konstantinos Fortounis scored from the penalty spot in the 81st minute to give his side the win that left them second in Group F on 10 points.
Real Madrid moved to the top of the Primera Liga after goals from Robinho and Alvaro Mejia clinched them a 2-0 win over Real Betis.
Betis fans bow to 'extraordinary' Messi after sublime hat-trick Lionel Messi is used to receiving adulation at the Nou Camp but on Sunday he got a rapturous reception from rival supporters for a majestic hat-trick in Barcelona's 4-1 win at Real Betis. Messi opened the scoring with an unstoppable free kick then combined with devastating effect with Luis Suarez to carve Betis open and score a second goal right before halftime. The Argentine saved his best goal for last, completing a victory which took Barca 10 points clear at the top of La Liga with a spectacular first-timed chip from the edge of the area after playing a one-two with Ivan Rakitic. Beaten Betis goalkeeper Pau Lopez looked on in awe while the home supporters, renowned for devotion to their side, rose to their feet, bowed down to Messi and chanted his name. "Even though the opposition concedes goals to Messi they can still enjoy this unbelievable era which he has created, they enjoy it too and today they recognised that," said Barca coach Ernesto Valverde. "It's significant that they gave him this ovation, it's an acknowledgement. Our rivals' supporters have to suffer him but they also appreciate him, because of the player he is. "Leo's performance was extraordinary, he got three goals but he could have got four, that shows you how good he is." Betis coach Quique Setien commended his supporters for their treatment of Messi while saying he felt honoured to work in the same era as the Argentine. "I really like the fact that our supporters showed their appreciation for a great player, today he really made us suffer," Setien said. "It's only fair and it's right to recognise this, our fans behaved in an extraordinary way and I'm very proud of them. I've seen some magnificent players do great things but no-one can match the things Messi has done over the last 12 years. "It's a privilege to have been able to start this era with him and see him in action every weekend." The five-times world player of the year moved on to 39 goals in all competitions this season with two months still left in the campaign. It was Messi's 51st career hat-trick but he could not recall receiving such applause from a rival team's fans. "I cannot remember anything like this before and I'm very grateful for the fans' reaction," Messi said. "I'm very grateful to them and happy for the victory, which was very important."
Real Betis came from a goal down to earn a 3-2 win at home to Real Zaragoza and stay in the hunt for a spot in the elite league.
Villarreal reclaimed fourth spot and a slim advantage in the race for a Champions League place with a 2-0 win at Malaga.
With the Catalans shorn of three of their top forwards, all eyes will be on Griezmann in his first appearance at the Camp Nou since his 120-million-euro ($133 million) move from Atletico Madrid, and not least because Barca need a win to kickstart their season.
La Liga has taken legal action against 10 people who allegedly threatened Ukraine international Roman Zozulya following his move to Rayo Vallecano last week, the Spanish league said in a statement on Thursday.
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