Real Betis secured a famous 2-0 victory at the Estadio Bernabeu as Real Madrid's disappointing season ended in defeat, while Barcelona drew with Eibar.
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Moron's 61st-minute strike and a second from former Real Madrid man Jese with a quarter of an hour remaining secured a deserved 2-0 win for the visitors on a day when Gareth Bale did not make it off the bench amid continuing speculation that he could be one of the casualties of a major overhaul by Zinedine Zidane this summer.
Zidane's men, who started the day 18 points adrift of champions Barcelona and eight behind derby rivals Atletico Madrid, were ultimately well beaten as their Liga campaign drew to a close with a third defeat in four games.
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How Betis beat Real
Madrid started in positive mood with Vinicius Junior very nearly catching keeper Pau Lopez in possession early on as he gave the home fans early encouragement.
However, Betis responded to pin their hosts back and keeper Keylor Navas was relieved to be in the right place at the right time to collect Giovani Lo Celso's 17th-minute header from Andres Guardado's free-kick.
Full-back Marcelo blazed high and wide from distance with 23 minutes gone, but went significantly closer two minutes later after Vinicius Junior and Luka Modric had worked a short corner with Zidane's men attempting to turn the tide.
But Navas had to tip over defender Marc Bartra's swerving 28th-minute drive as play switched rapidly from end to end.
Karim Benzema passed up a glorious opportunity to fire Madrid ahead when he latched on to Francisco Guerrero's ill judged 33rd-minute pass, only to hit the post with Pau saving Federico Valverde's well-struck follow-up.
However, Navas' fingertips came to the rescue at the other end seconds later after Lo Celso had attempted to lob him and the first half finished goalless.
Madrid stepped up a gear immediately after the restart, but were repeatedly unable to find a final ball of the required quality to hurt Betis.
Vinicius Junior was denied by Pau at his near post after turning brilliantly past Aissa Mandi into space with 57 minutes gone, and it was the visitors who took the lead within four minutes when Guardado found space on the left and his cross was perfectly weighted for Moron to fire home.
Navas had to save acrobatically from Lo Celso, who later curled a shot just wide of the far post with the keeper beaten as Betis looked to increase their advantage.
They did just that with 15 minutes remaining when substitute Jese was left unmarked in the box to latch on to Junior Firpo's pull-back and smash the ball into the roof of the net.
Barca held by Eibar
Lionel Messi struck twice for Barcelona as the title-winners were held to a 2-2 draw at Eibar in their final league game of the season.
Eibar took the lead against the La Liga champions through Marc Cucurella, before Messi fired back with two goals in two minutes to put Barcelona ahead.
In a game with plenty of chances, Eibar equalised on the stroke of half-time as Pablo De Blasis' incredible volley ensured the points were shared.
Barcelona wanted to end their league season on a high ahead of the Copa del Rey final against Valencia on May 25 as they look to complete the domestic double.
The game started openly, with both sides having shots saved in the opening five minutes at the Ipurua Municipal Stadium.
Barcelona came close to taking the lead when Jordi Alba sent a ball across the face of the Eibar goal, but no one was on the end of his cross.
Cucurella opened the scoring against his parent club as the 20-year-old got on the end of a Sergi Enrich pass, making the most of a Gerard Pique slip, before firing the ball underneath goalkeeper Jasper Cillessen.
Messi struck the equaliser after a ball from Arturo Vidal went through the defence and found the Argentinian, who took a touch before slotting home his 35th goal of the season.
The league's top goalscorer put his side in the lead just a minute later. After finding himself with large amounts of space on the halfway line, Messi ran forward before tapping the ball cleverly over the goalkeeper.
In a moment of concern for Barcelona, Nelson Semedo went down after a clash of heads and had to be replaced.
In the final moments of the first half, after Eibar forced Barcelona to play out from their box, the ball fell to De Blasis who volleyed expertly over Cillessen and into the net.
Barca's best chance of the second half came when Malcom crossed the ball in from the left but Pique in the centre of the box was unable to get close enough to tap it in.
Cucurella almost struck a winner for Eibar in the 80th minute but his effort from inside the box went just wide.
How the La Liga season ends
Champion Barcelona, Atletico, Madrid and fourth-place Valencia qualify for the Champions League. Getafe, Sevilla and Espanyol took the Europa League spots.
Girona, Huesca and Rayo Vallecano were relegated.