The 27-year-old, who had asked for a transfer in the close season and been linked in the media with a move to Juventus, was brought on for his first appearance in the league this season after 69 minutes.
With his side losing 2-1, he broke away and fired low across the goal to equalise two minutes later, celebrating with a huge roar to the small contingent of travelling fans.
Espanyol had stormed ahead in the first half with goals from Javi Lopez and Joan Verdu.
Bilbao started their fightback when Aritz Aduriz headed home in the 56th but, after Llorente levelled, Samuele Longo sprinted clear to net a third for the hosts with 10 minutes remaining.
Longo was sent off for a second bookable offence as he celebrated with Espanyol fans in the stands, and Aduriz bagged his second goal with a spectacular volley soon after to earn a point for Bilbao.
Marcelo Bielsa's side, who reached the finals of the Europa League and Copa del Rey last season, have now scored eight goals in their first four games and lie 13th in the standings with four points.
Barcelona top the table with the only perfect record in La Liga with 12 points after hammering Getafe 4-1 away on Saturday.