Arsene Wenger's side had drawn each of their previous three matches, but were able to ensure they remain three points behind leaders Chelsea by taking the three points
Alexis Sanchez continued his fine run of form by scoring twice in Arsenal's hard-fought 3-1 over a resilient Bournemouth at the Emirates Stadium on Sunday.
With Premier League leaders Chelsea, Manchester City and Liverpool all winning on Saturday 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 their 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 they edged ahead early in the second half thanks to Theo Walcott's header.
Bournemouth continued to cause problems and were adamant that they should have been awarded a second spot-kick late on as they desperately sought a leveller, but Sanchez broke the visitors' resolve for good in injury time with his seventh goal in his past eight games.
Arsenal stay fourth as a result of the victory, three points adrift of table-toppers Chelsea.
Bournemouth gave a good account of themselves during the opening exchanges, but they shot themselves in the foot in the 12th minute.
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.
And 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 they restored their 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 their improvement, Arsenal were 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 Benik Afobe from point-blank range.
But Bournemouth's desperation in attack was ultimately their undoing, as substitute Olivier Giroud cut back for Sanchez to finish off a rapid break to wrap things up in stoppage time.