After a turbulent final month of the campaign, Real looked determined to end the season on a positive note at the Bernabeu, as the club prepared to bid farewell to legendary right-back Carvajal.
The Spaniard, making his 451st and final appearance for Los Blancos, rolled back the years in the opening quarter-hour, curling a sublime pass into the path of Gonzalo Garcia, who took one touch to compose himself before finishing confidently beyond Alex Padilla.
That early breakthrough set the tone for the hosts, as Thiago Pitarch went close to doubling their lead, only for Padilla to thwart the midfielder’s effort.
At the other end, Athletic struggled to create anything of note in the final third, with Gorka Guruzeta directing a header wastefully off-target.
Having lost just one of the last seven meetings (W5, D1), Real added a second goal before half-time, as Jude Bellingham finished emphatically from Pitarch’s looped pass to score for the second time in three matches.
Undeterred, the visitors immediately pulled one back in first-half stoppage time through Guruzeta, who volleyed home from Inaki Williams’ inviting delivery to give Los Leones a lifeline.
Any hopes of a comeback were quickly extinguished by Los Blancos at the start of the second period, though, as Kylian Mbappe bagged his 25th LaLiga goal of the season via a clinical low finish from the edge of the box.
Athletic refused to lie down without a fight, and Ernesto Valverde’s men remained a threat, with Guruzeta and Adama Boiro both going close.
Those chances proved to be the visitors’ best opportunity of finding a route back into the contest, as Real cruised through the closing stages, adding a late fourth through substitute Brahim Diaz’s close-range finish.
And although Los Leones scored a consolation goal themselves after Urko Izeta’s towering header in the box, Real ended the season with a third straight league win, finishing eight points behind champions Barcelona.
As for Athletic, they missed out on a sixth consecutive top-half finish, having lost eight of their final 12 LaLiga matches.