Van Persie returned to the starting XI for the clash, having missed out for the midweek League Cup victory over Bolton Wanderers, whilst Park Chu-Young was on the bench following his winning goal in that tie.
Daniel Sturridge retained his place in manager Andre Villas-Boas' starting line-up following his extra-time winner against Everton in midweek, with Fernando Torres and Juan Mata completing the hosts' attacking front three.
The home side should have opened the scoring with just four minutes on the clock as Daniel Sturridge looked to play the ball across the goal to Fernando Torres who had a near enough open goalmouth, but the Englishman’s pass was poor and went straight into the arms of Wojciech Szczesny.
Arsenal then took their turn to squander an opportunity a few moments later, as Theo Walcott got behind Ashley Cole before delivering a great low cross, but Gervinho somehow missed the target from just a matter of yards.
Less than a minute later, Van Persie latched on to another Walcott cross, but the Dutchman failed to connect with the ball properly as his effort cleared the crossbar.
The Gunners were made to rue those misses moments later, as Frank Lampard gave the hosts the lead after 13 minutes, with Mata easily beat Andre Santos before putting in a brilliant delivery for the England midfielder to head past the helpless Szczesny.
Just shy of the half-hour mark, Sturridge should have added a second for Chelsea as Lampard played a remarkable ball over the top of the Arsenal defence for the young striker, but the finish was poor as the former Manchester City man scuffed his shot past the far post.
However, Arsenal got their equaliser in the 35th minute after a lovely move started by an excellent forward pass by Aaron Ramsey to Gervinho, with the Ivorian cheekily squaring the ball to Van Persie as Petr Cech approached, giving the striker an easy finish.
However, Wenger's side were only level for 10 minutes as John Terry looked to get his name in the headlines for the right reasons, bundling the ball past Szczesny after outmuscling centre-back Per Mertesacker at a corner to give Andre Villas-Boas' men the lead going in to half-time.
Van-tastic | RVP's first of three draw's Arsenal level at Stamford Bridge
The teams had not even been out for 40 seconds in the second half when Van Persie registered another shot at goal with Ramsey also going close also by the end of the minute, but it was Santos who drew the visitors level again.
Alex Song did brilliantly to pick out the unmarked Brazilian flying down the left-hand side, before the former Fenerbahce man stroked the ball past Cech with confidence.
Ten minutes into the second half, Arsenal got themselves into the lead for the first time of the day as Walcott marched through the Chelsea defence, before recovering well from a slip to produce a good strike that beat Cech.
With just under 20 minutes to go, Ramsey almost gave Arsenal a two-goal lead as the midfielder received a clever pass from Van Persie as he arrived late in the penalty area, but the Welshman’s shot was taken wide of the post thanks to a deflection.
Arsenal couldn’t keep hold of their lead however as Mata stepped up in the 80th minute, smashing a fantastic effort towards the far side of the goal, giving Szczesny no chance whatsoever.
Four minutes later however, suspect defending reared its head again with Florent Malouda playing a poor pass back to Terry, with the England captain's slip allowing Van Persie a free route to goal as the former Feyenoord man rounded Cech and slotted home to make it 4-3 to the visitors.
A minute into stoppage time, Arsenal secured the three points as Van Persie grabbed his hat-trick following a brilliant counter-attack, with Mikel Arteta picking out the Dutchman who smashed a swerving effort past Cech.
The three points see Arsenal join Tottenham on 16 points in the Premier League whilst Chelsea lose even more ground on the two Manchester clubs at the top of the table.
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