The Gunners could have moved level on points with Barclays Premier League top two Manchester City and Leicester but they were held to a 1-1 draw by a battling home side.
Mesut Ozil opened the scoring only for Lewis Grabban to equalise before half-time and Arsene Wenger's afternoon went from bad to worse as Sanchez hobbled off with a hamstring problem.
The Gunners boss had already lost defender Laurent Koscielny early on but Sanchez will be a huge miss if he is out for any length of time - with Wenger having admitted the former Barcelona man had felt a "little hamstring alarm" after the midweek win over Dinamo Zagreb.
Norwich battled back from a slow start and deserved a share of the spoils, although Alex Neil will feel his side made enough clear chances to win the game and pull themselves further clear of those scrapping at the foot of the table.
The visitors were able to give Aaron Ramsey his first start in almost six weeks and had Alex Oxlade-Chamberlain on the bench with both having recovered from hamstring issues.
But their wretched luck with injuries continued as Laurent Koscielny was forced off with less than 10 minutes gone after clashing with Lewis Grabban.
Both Ozil and Olivier Giroud spurned early chances for the visitors after a strong start from Wenger's side before t he visitors finally forced John Ruddy into action just before the half-hour mark as Sanchez stung the palms of the England goalkeeper with a rasping effort
The opener came just moments later as Gary O'Neil failed to control Ruddy's poor clearance and Sanchez again provided the assist for Ozil, who finished calmly.
Norwich finally began to stir and Robbie Brady tested Petr Cech as he curled in a right-footed effort which the Arsenal goalkeeper pushed away on the stretch.
The Republic of Ireland international was key to Norwich's equaliser from the next attack as he played the ball into the area where Grabban was too strong for substitute Gabriel and slotted home a simple finish on his first league start since the opening weekend of the season.
The Canaries should have been ahead at the interval but Wes Hoolahan somehow turned over Andre Wisdom's cross from three yards out as he ghosted in unmarked at the back-post.
The second-half started more evenly and neither side came close to taking the lead before Sanchez pulled up lame as Wenger's risk of playing his talisman backfired and he limped off to be replaced by Joel Campbell.
The loss of Sanchez seemed to knock Arsenal's confidence and Alex Neil's side set about taking full advantage.
Gabriel almost headed Martin Olsson's cross into his own goal only for Cech to make a stunning save, with Jonny Howson volleying the resulting corner over the crossbar.
Despite forcing the issue, Norwich could not find a late winner as Arsenal were left hanging on to take a point home on their plane back to London.
TWEET OF THE MATCH
"That Norwich halfwit should be sent off
Bloody stupid, dangerous shove on Alexis
#NCFCvAFC" - Piers Morgan @piersmorgan
PLAYER RATINGS
Norwich:
John Ruddy: 5
Andre Wisdom: 6
Ryan Bennett: 7
Sebastien Bassong: 6
Martin Olsson: 6
Jonny Howson: 6
Gary O'Neil: 5
Graham Dorrans: 5
Robbie Brady: 6
Wes Hoolahan: 5
Lewis Grabban: 7
Substitutes -
Nathan Redmond: 6
Vadis Odjidja-Ofoe: 6
Cameron Jerome:6
Arsenal -
Petr Cech: 6
Hector Bellerin: 6
Per Mertesacker: 6
Laurent Koscielny: 5
Nacho Monreal: 7
Mathieu Flamini: 6
Santi Cazorla: 6
Aaron Ramsey:6
Mesut Ozil: 8
Alexis Sanchez: 7
Olivier Giroud: 6
Substitutes -
Gabriel: 5
Joel Campbell: 6
Alex Oxlade-Chamberlain: 6
STAR PLAYER
Mesut Ozil: The World Cup winner has now been directly involved in 13 Arsenal goals in his last 12 Premier League appearances and he backed up what his manager has said about his improving finishing by tucking away the opener here
May not have hit the same heights as midweek but will be an even more crucial figure for the Gunners if Sanchez's injury proves serious.
MOMENT OF THE MATCH
Alexis Sanchez: Arsenal supporters would have been watching through their fingers as Sanchez pulled up after chasing a long ball
Arsenal manager Wenger revealed he had a worry over his key man but does not believe he can rest the Chilean
He might have no choice after this
Wes Hoolahan is lucky here as his glaring miss would have made moment of the match on most other days.
VIEW FROM THE BENCH
Arsenal have previous for failing to sustain their form and this result means the Gunners have won one of their last four league games
Wenger would have been pleased to welcome back Aaron Ramsey and Alex Oxlade-Chamberlain but to lose Laurent Koscielny and then Sanchez will leave him open to criticism
Alex Neil will be delighted his tweaks worked out as Lewis Grabban, starting his first Premier League game since the opening day defeat to Crystal Palace, came up with the goods.
MOAN OF THE MATCH
What was Wenger doing risking his best player when he had openly admitted he was not sure he was 100 per cent fit? Arsenal fans will be nervously waiting to hear the prognosis with a packed festive schedule and a Champions League decider with Olympiacos on the horizon.
WHO'S UP NEXT:
Watford v Norwich (Premier League - Saturday, December 5)
Arsenal v Sunderland (Premier League - Saturday, December 5)
Source : PA
Source: PA