The thumping derby win was a welcome tonic for the Catalan giants, who had been reeling from the shock €222 million departure of Neymar to Paris Saint-Germain in August.
The loss of the Brazil striker had sparked speculation Messi could follow him in leaving Camp Nou, with the Argentine yet to sign the new contract he agreed with the club in July.
Barca's sporting director Robert Fernandez has said he expects the Argentina international to put pen to paper "soon".
And Valverde is likely to be a relieved man when the 30-year-old formally commits his future to the Blaugrana.
"I am happy for Leo who scored three goals and is extraordinary," Valverde said at a post-match news conference.
"We pulled through a complicated game, because these kinds of matches are always tough.
"Despite the result, it wasn't easy. Espanyol fought very well, but we have great players like Messi, who scored three goals, and [Luis] Suarez, who gave everything like the rest of the team."