Da Costa posted a 1m33.752s aboard his Oreca 07 to move ahead of three cars running in the Hypercar class with 10 minutes left on the clock of the hour-and-a-half free practice session.
His time around the Algarve circuit put him just 0.029s ahead of the Alpine A480-Gibson LMP1 in which Nicolas Lapierre had earlier set a 1m33.781s.
The two Toyota Le Mans Hypercars were right behind the Alpine Oreca design as the top four cars were separated by just 0.055s.
Sebastien Buemi took third spot in the #8 GR010 Hybrid with a 1m33.800s, while Kamui Kobayashi was fractionally behind on a 1m33.807s
The new Glickenhaus 007 LMH ended up 12th with a 1m34.789s set by Richard Westbrook, which left the new Le Mans Hypercar just over a second off the pace.
Fifth fastest and second in LMP2 was Alex Brundle in the Inter Europol Oreca with a 1m33.979s lap, while Ben Hanley took sixth for DragonSpeed on 1m34.268s.
Paul di Resta and Giedo van der Garde were seventh and eighth in their respective United Autosports and Racing Team Nederland Orecas.
The second JOTA Oreca with Tom Blomqvist driving and the RealTeam Racing entry with Norman Nato at the wheel rounded out the top 10 overall.
Kevin Estre leapfrogged Porsche teammate Gianmaria Bruni at the end of the session to lead the way in GTE Pro.
The Frenchman’s 1m40.227s in the best of the factory Porsche 911 RSRs put him just over a tenth up on Bruni’s 1m40.389s.
Daniel Serra was three tenths back in the fastest of the AF Corse Ferrari 488 GTE Evos. The Brazilian’s 1m40.682s compared with 1m40.738s set by teammate James Calado in the sister car.
Matt Campbell led the way in GTE Am for the Dempsey-Proton Porsche squad. The Australian’s 1m41.202s put him just over two tenths up on Iron Lynx team boss Andrea Piccini.
Two further sessions of free practice on Saturday are followed by qualifying for Sunday’s second round of the 2021 WEC, the Portimao 8 Hours, at 6pm local time.