With Premier League leaders Chelsea, Manchester City and Liverpool all winning the day before, the onus was on the Gunners to respond on home turf.
Arsene Wenger's men did not have it all their own way, though, as Bournemouth - devoid of Arsenal-loanee Jack Wilshere - put in a good showing in north London.
It was a mistake of its own doing that saw Arsenal take the lead when Sanchez pounced on Steve Cook's error to break the deadlock, but Bournemouth's efforts were rewarded before half-time when Callum Wilson levelled from the penalty spot.
Arsenal produced a marked improvement after the break and afforded Bournemouth significantly fewer opportunities and it edged ahead early in the second half thanks to Theo Walcott's header.
Bournemouth continued to cause problems and was adamant that it should have been awarded a second spot-kick late on as they desperately sought a leveller, but Sanchez broke the visitor's resolve in injury time with his seventh goal in his past eight games.
Arsenal stays fourth as a result of the victory, three points adrift of table-topping Chelsea.
Cook's horrendous back pass went straight to Sanchez and the forward stroked his effort past the stranded Adam Federici from 18 yards.
Arsenal suffered a blow shortly after as Mathieu Debuchy - making his first Premier League start since November 8 2015 due to Hector Bellerin's injury absence - was replaced by Gabriel Paulista after picking up a hamstring problem.
Within eight minutes of that defensive reshuffle, Bournemouth restored parity.
Nacho Monreal was caught the wrong side of Wilson in the area and clumsily tripped him, with the striker dusting himself off to confidently dispatch the resulting penalty.
Bournemouth continued to show plenty of purpose in attack and tested Petr Cech twice in quick succession through Adam Smith and then Brad Smith.
But Arsenal's individual brilliance shone through on occasion and Sanchez smashed against the crossbar from a tight angle on the stroke of half-time.
Arsenal looked more solid in midfield and at the back after the break and restored its lead in the 53rd minute.
Mesut Ozil did well on the left, keeping the ball from his marker with clever footwork before delivering an inch-perfect cross to Monreal, who guided the ball into the danger zone where Walcott was on hand to nod in from close range.
Despite its improvement, Arsenal was lucky to stay ahead in the closing stages, as Monreal managed to avoid punishment for handling in the penalty area, before Cech denied former Arsenal trainee Benif Afobe from point-blank range.
But Bournemouth's desperation in attack was ultimately its undoing, as substitute Olivier Giroud cut back for Sanchez to finish off a rapid break to wrap things up in stoppage time.