The two London teams, who could now be on course for a semi-final meeting, will be relieved to have avoided the Italian favourites, Sampdoria, and the powerful Dutch side, Feyenoord.
Chelsea's opponents, Bruges, struggled past the League of Ireland side, Sligo Rovers, in the first round in September but, more impressively, dismissed the strong Greek team, Panathinaikos in the second round.
Arsenal and Chelsea were both originally drawn to play their home legs on 2 March, but Uefa have switched Chelsea's first game to Belgium, not wanting two European ties in the same city on the same night.
Arsenal, who won the Cup-Winners' Cup by beating Parma in Copenhagen in May, now face an Auxerre team who lost to Liverpool in the Uefa Cup three years ago. The Arsenal striker, Alan Smith, who scored the goal against Parma, said: "Teams generally like to have their away game first in Europe, so they know what they have to do on their own ground, but mainly it is a psychological thing.''
The Auxerre coach, Guy Roux, who has transformed the club from an amateur side to a First Division force in the 30 years he has been at the club, said: "There are no easy opponents at this stage of the competition and Arsenal are one of the strongest."
Sampdoria, the side for whom the England captain David Platt plays, face Porto, while Spain's Real Zaragoza meet Feyenoord, who ended Tottenham Hotspurs' hopes in the competition two years ago.
In the Uefa Cup draw, the three Italian teams left in the competition managed to avoid each other, as did the the three German clubs. Juventus play Eintracht Frankfurt, Lazio face Borussia Dortmund, while Parma are paired with Odense. The quarter-final line-up was completed with a match between Bayer Leverkusen and Nantes.
EUROPEAN CUP-WINNERS' CUP Quarter-final draw: Feyenoord v Real Zaragoza; Sampdoria v FC Porto; Arsenal v Auxerre; Club Bruges v Chelsea (First leg to be played on 2 March; second leg on 16 March).
UEFA Cup Quarter-final draw: Eintracht Frankfurt v Juventus; Lazio v Borussia Dortmund; Bayer Leverkusen v Nantes; Parma v Odense BK (First leg to be played on 28 February; second leg on 14 March).Reuse content