The competition's most successful club, with 13 wins, secured a last-16 trip to 2008 winners Portsmouth courtesy of a deserved triumph at the Vitality Stadium.
Bukayo Saka opened the scoring inside five minutes to cap a fine team move involving all 10 outfield players, with Eddie Nketiah doubling the visitors' lead before the break.
Arsenal made an early statement of intent with an excellent display of passing and movement that led to Saka thrashing home the opener from the corner of the six-yard box, Joe Willock and Gabriel Martinelli proving particularly influential in the build-up.
The industrious Willock was in the thick of the action again midway through the half, eluding Dan Gosling before calling goalkeeper Mark Travers into action.
Saka turned provider for Nketiah, the former Leeds United loanee rolling in from close range at the end of the opening goalscorer's low cross in the 26th minute.
Bournemouth, who ended a six-match winless run in the league with a 3-1 win over Brighton and Hove Albion last time out, looked bereft of ideas and spent much of the first half on the defensive, with Harry Wilson's strike providing the only meaningful work for Emiliano Martinez.
Arsenal lost Shkodran Mustafi to a serious-looking injury just past the hour mark and suffered a brief scare when Martinez flapped at a corner, the ball spinning wide off the head of the seemingly oblivious Jack Simpson.
Arteta's side appeared to be cruising until Surridge halved the deficit in stoppage time from fellow substitute Simon Francis' probing ball, but it was too little too late and the Spaniard will be hopeful of adding to the two FA Cups he won as a player with the Gunners.
What does it mean? Arteta eyes season's silver lining
While a top-four finish in the Premier League may not be beyond Arsenal's reach, it is a significant stretch for a side whose top-flight win tally of six matches Bournemouth's exploits this season. That leaves the FA Cup as a sizeable consolation prize for a club with a proud history in this competition.
Lively Saka catches the eye
Saka's impact on the game is evident from his goal and assist, but the 18-year-old's all-round display was just as noteworthy, with an 87.5 per cent passing accuracy, five recoveries and two chances created.
Howe bad was that?
Cherries boss Howe made six changes from the victory against Brighton and, with top-flight safety his primary objective, you can understand why. But the side he put out looked lost, producing a disjointed and tactically inept display that saw the tie put beyond their reach before the half-hour mark, with Surridge's late strike failing to paper over the cracks.
Arsenal head to Turf Moor on Sunday to take on a Burnley side who sit level with them on 30 points, while Bournemouth welcome Aston Villa for a crunch six-pointer on Saturday.