The England winger claimed Real's first goal, after a brilliant jinking run, as John Toshack's side twice came from behind. Guti, from a cross by the substitute Nicolas Anelka, scored their second, while Bilbao's goals came from a Julen Guerrero free-kick and a Geremi Njitap own goal.
Real's city rivals, Atletico, are bottom of the table with no points after a 2-1 home defeat to Celta Vigo last night.
Valencia, who entertain Rangers in the Champions' League tomorrow, have also lost all three games so far. They went down 2-0 at home to Alaves after letting in two goals in the last 10 minutes and having Joachim Bjorklund, the former Rangers defender, sent off.
A full house of 98,000 at the Nou Camp on Saturday saw Barcelona win the first Catalan capital derby of the season with a comfortable 3-0 home victory over Espanyol.
All three goals came in the second half, after an ugly clash between Barcelona's Luis Figo and Mauricio Pochettino. The Portuguese midfielder lashed out after the last of several late tackles from his marker, and both men were sent off in first-half injury-time.
After the break the Brazilian playmaker Rivaldo scored twice and Patrick Kluivert grabbed his first league goal of the season as Barcelona strolled to their third win out of three. The only other side with maximum points are the newly-promoted Madrid side Rayo Vallecano, who beat Real Sociedad 2-1 yesterday.
Roy Prager hit a hat-trick to help Hamburg thrash Hertha Berlin 5-1 on Saturday and stay top of the German Bundesliga on goal difference ahead of Bayer Leverkusen, who came from behind to win 3-2 at home to Schalke 04.
The champions, Bayern Munich, who have made a poor start to the season, welcomed back their captain Stefan Effenberg from a long lay-off but still failed to shine in a 2-1 win at MSV Duisburg. Michael Tarnat and Thomas Linke were on target for the Bavarians.
The most bizarre Bundesliga game was on Friday at Hansa Rostock, where the visitors, Ulm, finished with only seven men on the pitch after four red cards. They equalised at 1-1 in the 79th minute when they had eight players, and lost to an injury-time Rostock goal moments after the final dismissal.
In Italy, Fiorentina, who play host to Arsenal tomorrow in their first European Cup campaign for 30 years, led 2-1 for much of their game at newly promoted Reggina but conceded a late goal and had to settle for a 2-2 draw. Aldo Firicano and Jorg Heinrich were on target for the Florence club.
Neither Christian Vieri nor Ronaldo could provide the answer for Internazionale in a 0-0 draw at Roma last night.
Vieri, held out of the starting line-up with a nagging hamstring problem, replaced an ineffective Ronaldo midway through the second half, but had no more success at threatening to score. It was Roma, with Francesco Totti and their close-season acquisition Vincenzo Montella leading the attack, who created the better chances, but the Inter goalkeeper, Angelo Peruzzi, was equal to the task each time. The result meant that no club in Serie A have won their first two games of the new season.
Croatia Zagreb, Manchester United's Champions' League opponents tomorrow, won 4-0 at Istra on Saturday to claim second place in the Croatian premier league.Reuse content