The Gunners needed to beat David Moyes’ men to take the battle to finish in the top four of the Premier League down to the last day of the campaign.
They made heavy weather of their task, with Sanchez scoring a close-range double inside the final 17 minutes to secure the points with a 2-0 win.
Under-fire Arsenal boss Arsene Wenger has never finished outside of the top four in a full season at the helm but despite this win, and even though it remains mathematically possible, the club look set for a Europa League campaign next year.
Manchester City’s victory over West Brom means Arsenal need to beat Everton on Sunday and hope Liverpool slip up at home to Middlesbrough to gatecrash the top four.
Wenger’s own position still remains a mystery as he has yet to announce if he will sign a new deal to remain at the club beyond the FA Cup final meeting with Chelsea in two weeks’ time.
A lot of the home fans who stayed away from this clash did so in protest against Wenger remaining in charge and those who opted to attend were left short-changed.
The official attendance of 59,510 was certainly wide of the mark, with Arsenal recording tickets sold rather than the figure who attended.
Arsenal thought they were ahead early on but saw a goal correctly chalked off as a Sanchez shot hit Giroud on the arm as it flew past Jordan Pickford in the visitors’ goal.
Moments later the young Black Cats goalkeeper, who has been linked with a move to the Emirates, did well to keep out a genuine Giroud strike as the France striker found himself free in front of goal.
Despite camping out in the Sunderland half, Wenger’s side had few clear-cut chances after his early save from Giroud and it was his counterpart Petr Cech who had to dive low to stop Didier N’Gong scoring against the run of play.
Sanchez finally opened the scoring soon when he converted an easy chance in front of goal as Mesut Ozil crossed from a pinpoint Xhaka pass.
The Chilean had been on the periphery but hit the crossbar with a cheeky lobbed effort, with Pickford again on hand to deny Xhaka and Alex Iwobi in spectacular fashion.
Pickford was powerless to prevent Sanchez doubling the lead with nine minutes remaining, parrying Giroud’s flick into the path of the former Barcelona man, who headed in to seal the victory.
Mustafi had a late header brilliantly turned on to the bar by Pickford.