Sergio Aguero also came on as a late substitute to make his first appearance for Barcelona.
The hosts got off the worst possible start following Jose Gaya’s superb strike from long range with just four minutes on the clock.
But Ansu Fati hit back for Barcelona nine minutes later before Memphis Depay scored from the spot as the first half was drawing to a close.
Former Liverpool forward Philippe Coutinho then made sure of the victory with five minutes remaining.
Sergio Aguero also came on as a late substitute to make his first appearance for Koeman’s side since leaving Manchester City.
The win moves Barca up to seventh, five points behind leaders Real Sociedad.
Rafa Mir’s second-half strike secured a 1-0 win for Sevilla at Celta Vigo, while Rayo Vallecano saw off Elche 2-1.
Ezequiel Avila scored with four minutes left as Osasuna won 2-1 at Villarreal.
In Germany, Bayern Munich scored five times in 37 scintillating first-half minutes to thrash Bayer Leverkusen and take top spot in the Bundesliga.
Robert Lewandowski opened the scoring with a cute backheel after just three minutes before adding his second on the half-hour mark.
Moments later, Thomas Muller secured his side’s third with Serge Gnabry netting goals four and five after 35 and 37 minutes.
Patrick Schick pulled one back for the shell-shocked hosts in the second half.
In the day’s other fixture, Augsburg and Arminia Bielefeld shared the spoils with a 1-1 draw.
Victor Osimhen scored a late header as Napoli retained their 100 per cent start to Serie A.
The Nigeria striker netted with just nine minutes remaining to see off a spirited Torino and move two points clear of AC Milan.
Moise Kean’s header was enough for Juventus to secure a 1-0 win against Jose Mourinho’s Roma.
The on-loan Everton forward inadvertently deflected in Rodrigo Bentancur’s header after 16 minutes to earn the Old Lady a fourth consecutive league win.
Jordan Veretout missed a chance to draw Roma level when his penalty – after Tammy Abraham was fouled before subsequently scoring after the referee had blown his whistle – was saved by Wojciech Szczesny.
Roma are nine points behind Napoli in fourth, while Juventus move up to seventh.
Elsewhere, Josip Ilicic scored twice as Atalanta won 4-1 at Empoli, while 10-man Udinese fought back from behind to draw 1-1 with Bologna.
In Ligue 1, former Arsenal youth striker Stephy Mavididi scored the winner as second-place Lens suffered a 1-0 defeat against Montpellier.
Mavididi found the back of the net two minutes into the second half as Lens were cast nine points adrift of leaders Paris Saint-Germain.
Troyes’ Mama Balde condemned Nice to a 1-0 defeat to leave them in third, while Marseille ran out 3-1 winners in the evening fixture against Lorient.