The striker calmly rolled in his first after running clear, before tapping in a second from close range.
Jordi Alba smashed home a third and David Lopez stuck for Espanyol before Lionel Messi added a fourth late on.
Barca are now three points behind leaders Real Madrid, who have played one game fewer after being crowned Club World Cup champions earlier on Sunday.
Barcelona forward Messi produced two moments of brilliance in the second half to give Luis Enrique’s side a comfortable victory at the Nou Camp.
The Argentine danced past four Espanyol defenders for Suarez’s second, with the Uruguayan quickest to react following Espanyol goalkeeper Roberto’s fumble.
A minute later, Messi beat the same number of players before laying the ball into the path of Alba to score.
Messi completed his masterclass performance with a well-deserved goal, poking in through the legs of Roberto from Suarez’s lobbed pass.