Having fallen to back-to-back defeats in their last two matches, the Gunners picked up their first points since the opening day of the season as a brace from Danny Welbeck either side of an Alexandre Lacazette strike saw them ease past the winless Cherries.
Bournemouth have now lost their opening four games of the season and remain in the relegation zone as their search for their first points of the campaign drags on.
Manager Arsene Wenger would have been demanding a quick start from his side following their 4-0 humiliation at the hands of Liverpool before the international break, and the players responded with the opening goal after just six minutes.
It was a sign of things to come as Arsenal dominated the first half, and Begovic was by far the busier of the two goalkeepers with saves to deny Granit Xhaka and Mesut Ozil, the latter of which saw him palm a goalbound free kick away with a fine one-handed stop.
Xhaka missed the target with another effort before the hosts deservedly doubled their lead through Lacazette, who collected Welbeck's pass before curling an accomplished finish into the top corner from 25 yards.
The goal saw Lacazette become just the second player to score in his first two Premier League home games for Arsenal, and he should have added a third 10 minutes later following an error from Begovic.
The Bournemouth goalkeeper took too much time over his clearance and was charged down by Welbeck, but atoned for his mistake by quickly reacting to smother Lacazette's follow-up effort from inside the area.
Having offered no threat in the opening 45 minutes, Bournemouth almost pulled a goal back within three minutes of the restart when Jermain Defoe got on the end of Jordon Ibe's cross but saw his downward header come back off the post before failing to react in time to convert the rebound.
It proved to be a pivotal moment in the match as Arsenal scored a third shortly afterwards to take the game beyond the Cherries when Welbeck doubled his own personal tally with a low finish into the bottom corner after being played through by Ramsey.
Ozil and Ramsey both threatened to add a fourth before Welbeck came within inches of completing his hat-trick in style, bursting into the box before lifting an audacious chipped finish just past the far post.
Wenger turned to contract rebel Alexis Sanchez in his bid to add a fourth goal, and the Chilean set up fellow substitute Olivier Giroud moments after his introduction as the French striker forced another save from Begovic.
It took until the 89th minute for Bournemouth to finally call Petr Cech into action at the other end, and even then it was a routine stop for the Arsenal keeper as he tipped Joshua King's looping header over the crossbar.
Sanchez drew one more save from Begovic before Bournemouth were finally put out of their misery, with the defeat seeing them become just the ninth team to lose their opening four fixtures of a new Premier League season.
Arsenal, meanwhile, climb into the top half as a result of the victory and have now won their last seven Premier League matches at the Emirates Stadium.