After spurning several golden opportunities, Arsenal's persistence paid off when Bukayo Saka showed great composure to slot home his penalty on the stroke of half time.
Leandro Trossard doubled the home side's advantage in the 70th minute, guiding the ball past Mark Travers to register his 10th goal of the Premier League season.
Mikel Arteta's side were given a slight scare when Antoine Semenyo found the net, before the Ghanian's strike was disallowed for a foul against David Raya.
Despite seeing Gabriel Magalhaes's fierce strike ruled out, the Gunners eventually added a third thanks to Declan Rice's 97th-minute shot.
A fourth consecutive victory moves Arsenal four points clear of Manchester City, who will have the chance to respond in Saturday's evening kick-off against Wolverhampton Wanderers.
Saka puts Arsenal ahead
Travers came into the contest off the back of consecutive clean sheets after replacing Neto in the Bournemouth side earlier this month.
The Republic of Ireland international was soon made aware he would need to produce something special to record a third successive shutout, as Arsenal launched an onslaught in the opening half-hour.
Travers got down low to his right to keep Kai Havertz at bay, before he was required to make another strong stop to thwart William Saliba's left-foot shot.
Saka was the next Arsenal player to try his luck, but the England winger could find no way past Travers at the near post.
The 24-year-old goalkeeper experienced some brief respite, before he was forced to make a diving stop to push Thomas Partey's low strike away from danger.
Bournemouth looked set to scrape through to half time when Rice thumped the ball wide from Havertz's neat knockdown.
However, Arsenal were given the chance to break the deadlock after Travers was adjudged to have brought down Havertz in the box.
Saka shouldered the responsibility to calmly roll the ball home and register his 16th Premier League goal of the season.
Trossard, Rice put the game beyond doubt
Travers was unable to stop Saka from 12 yards, but he made an important save to prevent the England international from grabbing a second in after the restart.
Bournemouth started to show signs of improvement, with Dominic Solanke forcing Raya into his first piece of meaningful action.
However, just as the Cherries began to grow in confidence, Arsenal produced a sucker punch to double their advantage with 20 minutes left to play.
Rice set the ball out to Trossard for the Belgium international to sweep past Travers and score his third goal in four matches.
After being left with a mountain to climb, Andoni Iraola's side thought they had quickly halved the deficit, only for Semenyo's strike to be chalked off for what referee David Coote deemed to be a foul on Raya by Solanke.
Gabriel looked to have netted Arsenal's third when he sent a thunderous strike out into the top corner, but his unstoppable effort was ultimately disallowed for offside.
Despite that minor setback, the Gunners still managed to grab a third goal in the seventh minute of stoppage time when Rice powered his near-post shot past Travers.
The commanding win will pile the pressure on Man City, who sit four points adrift with two games in hand, including Saturday's home clash with Wolves.
Arsenal will now turn their focus to a mouthwatering clash with Manchester United, while 10th-placed Bournemouth will begin preparations for a home game against Brentford.