Suarez scores a brace but Messi injured in win over Las Palmas; Real Madrid held while Villareal go top of La Liga.
Champions Barcelona suffered their second defeat in three matches in La Liga with Michael Krohn-Dehli scoring and setting up another goal in a 2-1 victory for Sevilla.
Denmark international Krohn-Dehli tucked the ball home after 52 minutes following a pass from Kevin Gameiro to give Sevilla the lead and then provided a cross for Vicente Iborra to nod in six minutes later.
5 Nolito (Celta Vigo)
Karim Benzema (Real Madrid)
Cristiano Ronaldo (Real Madrid)
4 Iago Aspas (Celta Vigo)
Lucas (Deportivo Coruna)
Imanol Agirretxe (Real Sociedad)
Barca, with Lionel Messi absent through injury, hit the woodwork on four occasions, but luck was not on their side as they lost their second consecutive away game following their 4-1 defeat by Celta Vigo.
For Sevilla, victory was some revenge for their European Super Cup loss in August.
Neymar struck the post with a free-kick after 23 minutes, with the ball then striking the woodwork again after rebounding off keeper Sergio Rico, and running across the goal line just out of the reach of Gerard Pique.
Luis Suarez also fired against the crossbar with a curling effort from outside the area before half time but Barca’s defending was poor and they paid the price after the break.
Neymar scored from the penalty spot on his 100th appearance after a 74th-minute handball from Benoit Tremoulinas and then substitute Sandro Ramirez was denied by the woodwork with 10 minutes to go.
Barca have 15 points from seven games, a point behind leaders Villarreal who are away to Levante on Sunday.
Fourth-placed Real Madrid, on 14 points, take on Atletico Madrid who are one place and two points below them also on Sunday.