The Spain international then poured further misery on Arsene Wenger's side, guiding home Branislav Ivanovic's cross at the near post for the only goal of the game.
Victory ensures the Gunners finish top of Group B with one game - a trip to Dundalk next Thursday - left to play and head into Sunday's north London derby against Tottenham, live on Sky Sports, on a high.
Liverpool were without first-choice keeper Alisson because of illness but stand-in Caoimhin Kelleher made a fine save to deny David Datro Fofana an equaliser when the score was 2-1.
Manchester City beat Manchester United 2-1 to extend their perfect start to the Premier League season and end that of their local rivals in an entertaining and action-packed encounter at Old Trafford.
In a chaotic finale to a largely scrappy game, United first thought they had snatched three points only to see Alejandro Garnacho's 88th-minute finish ruled out for offside by the video assistant referee with the score 1-1.
Villa sealed a comfortable 4-1 victory over the Dutch outfit at AFAS Stadion and they are second in the table at the halfway point of the group stage, level on points with leaders Legia Warsaw AZ reached the semi-finals of the Conference League