One of former ONCE team leader Joseba Beloki's main lieutenants, Azevedo will very likely play a major role at the Tour de France and other top stage races. With proven success in recent Grand Tours—fifth at the 2001 Tour of Italy and sixth at the 2002 Tour de France—and other premier events, including fifth at both the 2002 Fleche Wallonne and 2001 Paris-Nice, Azevedo should find his place among the other climbers on the USPS team.
Azevedo closed out a successful three year run with ONCE in 2003 with a stage win and second overall finish at the Tour of Germany, a 10th place effort at Semana Catalana and a 26th place finish at the Tour de France. Azevedo started the Vuelta and was part of ONCE's winning team time trial effort in the opening stage but was unable to finish stage 10 and dropped out of the race. In Germany, Azevedo won the most difficult stage of the race to Feldberg in grand fashion as he and teammate Igor Gonzalez de Galdeano crossed together in full celebration, leading a top four sweep by ONCE. Azevedo also raced in the U.S. following the Tour at the New York City Cycling Championship.
In 2002, Azevedo scored a fifth place finish at Fleche Wallonne, coming to the line in a group of nine riders vying for third place, some six seconds back of winner Mario Aerts. Following La Fleche, Azevedo came through with third place efforts at the Midi Libre (won by Armstrong) and the Bicicleta Vasca (keyed by a fourth place finish in the time trial stage). In his first Tour de France, Azevedo finished an impressive sixth overall—one of three ONCE riders in the top six—to help lead the Spanish team to victory in the team classification. While working for team leader Beloki, Azevedo was still able to finish strongly in all the key stages, including eighth in the stage to La Plagne, 10th at La Mongie, 12th on Mount Ventoux and 14th to Les Deux Alpes.
The Portuguese National time trial champion in 1996 and 1997, Azevedo's 2001 was highlighted by his fifth at the Giro (holding a top three spot from stage eight through 12 and second place on stage 17), fifth at Paris-Nice (second to Sisteron, third in the time trial to Col d'Eze), seventh at the Tour of Valencia and eighth at the Tour of the Basque Country.