Sánchez's first-half strike in added time settled an edgy contest as Arsène Wenger's side navigated their way through a seventh group stage qualifier in the last nine seasons.
Like the first leg, Arsenal ended the match with 10 men after Mathieu Debuchy was sent off for two bookable challenges but Laurent Koscielny and Per Mertesacker expertly marshalled a watertight Gunners defence.
Arsenal rode their luck in the first-half when separate Debuchy and Jack Wilshere challenges inside the area were overlooked by the referee, however there were scant opportunities for the Beşiktaş frontman Demba Ba.
Sánchez struck on the stroke of half-time when his shot crept underneath the feeble Tolga Zengin to settle the tie. Ba was gifted a presentable chance in the 88th minute but failed to make contact from the cross.