They surpassed the previous record of 263-5, set by Royal Challengers Bangalore in 2013
Suarez latched on to a pass from Lionel Messi to dribble round the home side's goalkeeper and slot into an empty net in the 16th minute, while Messi hit the woodwork later in the game as did Eibar's Fabian Orellana, who was sent off with 25 minutes remaining for a second booking.
Eibar coach Jose Luis Mendilibar was also dismissed later for angry protests against referee Alejandro Hernandez Hernandez.