Liverpool did enough to build themselves a solid lead for Tuesday's return game in Spain where they will be confident of finishing off the job.
Villarreal became the last team to reach the Champions League knockout stages.
Depay calmly went around goalkeeper Geronimo Rulli to score a dramatic 88th-minute goal that restored their lead.
Kepa famously refused to be replaced by former Chelsea boss Maurizio Sarri in a League Cup final defeat to Manchester City two years ago.
Champions League winners Chelsea will face Europa League holders Villarreal in the Aug. 11 clash.
Rashford says he was racially abused on social media following Manchester United's defeat in the Europa League final.
Manchester United's gut-wrenching defeat on penalties by Villarreal in the Europa League final re-emphasised the magnitude of the rebuilding task for Solskjaer.
Arsenal found themselves against the ropes when they trailed 2-0 following goals by Manu Trigueros and Raul Albiol before having Dani Ceballos sent off.
Sitting 10th in the Premier League, the Gunners are on course for their worst league finish since 1995.
Villarreal have announced that long-serving midfielders Bruno Soriano and Santi Cazorla are leaving the club after playing in the final La Liga game of the season at home to Eibar on Sunday.
Iago Aspas struck twice on his return from injury to earn Celta Vigo a crucial 3-2 win over Villarreal on Saturday, leaving them one point and one position behind their 17th-placed opponents in the La Liga standings.
Real Madrid coach Santiago Solari is pleased with the progress they have made despite the revival being punctured by Thursday's 2-2 La Liga draw at Villarreal, emphasising the difficult situation he inherited when he took charge in October.
The 27-year-old will join manager Javier Calleja's side as they look to improve upon their fifth-placed finish last season.
Real Madrid are suffering this season in the league. They are 16 points behind the leaders, Barcelona.
Villarreal was the better side in the whole game, and their coach, Javier Calleja, counted on counter attacks to break Real Madrid’s defense
Cristiano Ronaldo and Gareth Bale will reappear in the Real Madrid line-up for the league match against Villarreal on Sunday.
AC Milan, Villarreal and Nice reached the knockout stage of the Europa League after posting wins on Thursday in the penultimate round of the group stage.