Van Persie set Arsenal on the way in the 65th minute before Edu struck in injury time to leave Albion a point away from safety with just two games remaining and a trip to Manchester United to come on Saturday.
The decision by Bryan Robson to flood the midfield with five men and operate with just former Gunner Kevin Campbell as a lone striker was a tactic that Arsenal struggled to get to grips with in the opening half.
It took them more then half an hour to create a chance as Albion made all the early running.
But, while Albion enjoyed plenty of possession, they struggled to force goalkeeper Jens Lehmann into a save.
Zoltan Gera flashed an acrobatic overhead volley just wide in the eighth minute before Ronnie Wallwork saw his drive blocked and Kieran Richardson curled a free-kick just wide.
As the game progressed and Arsenal started to adjust to Albion's change in tactics, they started to gradually work their way into the game.
Skipper Patrick Vieira set the tone for an impressive final ten-minute spell by Arsenal in the first half when he collected a pass from Robert Pires and fired just over the bar from 20 yards in the 35th minute.
Vieira then saw another drive blocked by Wallwork four minutes later before Albion goalkeeper Russell Hoult came to the rescue of his side with a stunning save.
Albion's defence was dissected by a 39th minute through ball from Lauren that was collected by Van Persie, who only had Hoult to beat from ten yards but the keeper spread himself to beat away the shot from the Dutchman.
Having successfully managed to ride their luck, Albion were themselves just inches away from breaking the deadlock on the stroke of half-time.
Wallwork raced on to a pass from Gera and fired in a rising drive from the edge of the area that whistled just over the bar.
Just as in the first-half, Albion really built up a head of steam at the beginning of the second period and were unfortunate not to make that pressure tell.
Lauren had to be well placed to block a 50th minute drive from Paul Robinson and six minutes later, Arsenal were relieved to see Gilberto Silva deflect a header from Neil Clement wide.
It proved to be a vital intervention by Silva as Arsenal took full advantage of the let-off to stun Albion when Van Persie struck with a neat strike to give Arsenal the lead.
The hard-working Vieira started the move when he picked out Jose Antonio Reyes.
Reyes then fed the ball into the path of Van Persie who drilled a low drive past Hoult from the edge of the area.
Reyes should then have doubled Arsenal's advantage after 72 minutes when he waltzed round Hoult to be faced with an empty net, but Clement made up ground to steer his shot off the line.
Predictably, Albion then made a late charge with Richardson stinging the fingers of Lehmann from 20 yards in the 76th minute.
In a last desperate throw of the dice, Robson then threw on three strikers, Kanu, Robert Earnshaw and Geoff Horsfield, in what proved to be a fruitless attempt to salvage something from the game.
With Albion committed to attack, it was no surprise that they left gaps in defence in the closing stages.
Those were fully exposed by a neat pass by Dennis Bergkamp, who set up fellow substitute Edu to loft the ball past Hoult and set the seal on the win.