After a drab first-half display the Gunners sprang in to life in the second and opened the scoring when Van Persie beat Jussi Jaaskelainen at his near post just 40 seconds after the break.
David Wheater then saw red for pulling back Theo Walcott when he was through on goal before Van Persie brought up his ton with a clever flick and Alex Song struck near the death to complete the morale-boosting win.
The win, Arsenal's second of the season, moved them away from the relegation zone and ended a difficult seven days for the club.
After having come from behind to beat League Two Shrewsbury in midweek, and seeing Arsenal chief executive Ivan Gazidis give him his full backing, manager Arsene Wenger will sleep a lot easier than he did last Saturday when his team crumbled against lowly Blackburn.
There were signs at the start of the game that Arsenal were going to put in another poor performance after they started nervously in front of their demanding home support.
Bolton, who left Gary Cahill at home and Kevin Davies on the bench, almost took the lead in the third minute when Wojciech Szczesny pulled off a brilliant save to deny Darren Pratley from six yards.
Wheater then just missed a glancing header from eight yards before Arsenal finally woke up.
Mikel Arteta was given space in midfield and sent Gervinho through with a brilliant slide-rule pass, but the striker's first touch was poor and Jussi Jaaskelainen came out to smother.
Van Persie then came within inches of handing Arsenal the lead in the 10th minute when he picked up Arteta's short pass from a free-kick and lashed a wicked curling shot just wide of the Bolton goal.
Davies' rare spell among the substitutes did not last long as he came on in the 20th minute to replace David Ngog, who had to come off after clashing heads with Laurent Koscielny.
The disjointed passing and failure to pressure Bolton while they were on the ball led to growing frustration among the home support.
Gervinho wasted a good chance when he blasted over from 20 yards before Laurent Koscielny's overhead kick dribbled wide.
Two blocks at the end of the half from Paul Robinson and Wheater ensured that Bolton's defence remained unbreached by the half-time whistle, which was greeted by some boos from the home support.
Arsenal were a completely different team in the second half.
They came out flying after the break and took just 40 seconds to score.
Aaron Ramsey strode forward with the ball and played in Van Persie, the Dutchman held off Fabrice Muamba's strong challenge and fired low past Jaaskelainen at his near post to register his 99th goal for Arsenal
The goal galvanised Arsenal and Jaaskelainen saved well from Van Persie after the striker had ventured in to the box.
Bolton were on the rack and their task was made much harder in the 55th minute when Wheater pulled Walcott back just outside the box after being played through by Ramsey.
With the England winger being through on goal with only Jaaskelainen to beat, referee Mark Clattenburg had no choice but to send Wheater off.
The Gunners were by now were clearly in the ascendancy and Arteta came close to doubling their lead with a side-footed effort that crept just wide.
Jaaskelainen blocked well from the Spaniard before Bolton hit back on the counter with a rare attack. Chris Eagles raced clear after being played in by Martin Petrov but the former Manchester United man wasted his golden opportunity by hitting a weak shot in to Szczesny's arms.
Jaaskelainen had to scamper across his line to deny Van Persie and was then tested by a powerful Laurent Koscielny volley.
With 20 minutes left Van Persie brought up his Arsenal ton and gave the Gunners the two-goal cushion they needed to feel safe.
Walcott beat his marker down the right and crossed low for the Dutchman, who turned home with a cheeky flick which beat Jaaskelainen from five yards.
Walcott should have made it 3-0 when he raced through after a brilliant ball by Song but Jaaskelainen palmed the winger's weak effort wide.
Only a crucial block from Zat Knight prevented Van Persie from scoring his hat trick with six minutes left before the former Feyenoord man came off to a standing ovation when he was replaced by Marouane Chamakh.
The rout was completed two minutes from time when Bacary Sagna's pull-back was met by Song, who cracked a wonderful curler past Jaaskelainen to make it 3-0.
With victory now guaranteed, the home support broke in to a chant of "there's only one Arsene Wenger" but there were worrying scenes for the Arsenal boss and the Gunners support at the end of the match when Walcott pulled up and hobbled off through injury.