With Spurs defeating Fulham 2-0 at White Hart Lane, Arsenal needed to take all three points to end a testing season with guaranteed Champions League football and they had to come from 2-1 down to do it.
The on-loan Yossi Benayoun marked what is expected to be his final appearance for Arsenal with a goal after only four minutes, the midfielder exploiting a terrible mistake from Fulop as the goalkeeper raced to the edge of the box and allowed the ball to slip away from him.
Shane Long looked to be offside when equalising after 12 minutes and Graham Dorrans then exploited some poor defending to give West Brom a shock lead in what was Roy Hodgson's final game before taking charge of England.
Andre Santos levelled after 30 minutes - his shot creeping past a culpable Fulop - before Arsenal exploited another gaffe from the Baggies keeper after 55 minutes. Fulop tried to punch clear from a corner yet palmed the ball straight at Laurent Koscielny, who turned it home from close range.
Arsenal looked vulnerable throughout - fittingly given the unconvincing nature of their season as a whole - yet held on for a win that secures their place in the Champions League for a 15th season in succession.
While Arsenal made three changes from the side that drew 3-3 against Norwich - Santos, Francis Coquelin and Carl Jenkinson coming in for Kieran Gibbs, Aaron Ramsey and the injured Bacary Sagna - West Brom were forced into a key change of their own as Ben Foster succumbed to injury and Fulop was required to make his first appearance since September.
After only four minutes, Fulop's rustiness was exposed as he was guilty of a calamitous error to allow Arsenal to take the lead.
The goalkeeper came off his line to intercept a loose ball, but when reaching the edge of the box appeared to be caught in two minds as to whether to pick it up or boot clear. He did neither, executing a spontaneous stepover that allowed Benayoun to nip in, take the ball away and tap home.
However, for the second week in succession an early Benayoun goal was followed by an Arsenal collapse as West Brom edged in front thanks to two goals in three minutes.
Arsenal were first carved open in midfield as James Morrison was allowed to saunter forward 20 yards before releasing Long with a neat through ball. Long was clearly offside when the pass was made but was allowed to advance on goal and punish a hesitant Wojciech Szczesny when slotting the ball past the static keeper.
Arsene Wenger was furious with the failure of the officials to spot the offside yet was soon turning his ire towards his own defenders. Morrison, again, chipped a pass into Dorrans 25 yards from goal and he escaped the clutches of a culpable Koscielny with a neat header before thumping home a volley from the edge of the box to make it 2-1.
It was a moment typical of a season that has seen Arsenal repeatedly fail to impress in defence. Indeed, it was the 49th league goal they had conceded - the club's worst tally since the 1994-95 campaign when they last finished behind Tottenham Hotspur in the league.
In attack they were unable to coax the best out of Premier League top scorer Robin van Persie in the first half, though the Dutchman was only denied by a superb block from Jonas Olsson just inside the box after collecting a pass from Tomas Rosicky.
Benayoun, playing his last game before the expiration of his loan from Chelsea, also drove a volley off target while Gervinho threatened in bursts, the right-sided forward finding the side netting after also forcing Fulop into a smart low save.
However, by that point the Hungarian had already been guilty of another poor piece of goalkeeping. Santos, typically in an advanced role despite his nominal status as a defender, snuck away from two markers before unleashing a firm shot with his left boot. It flew towards the near corner but Fulop should have done better than merely palming it into the side of the net for 2-2.
Fulop was again at fault when Arsenal stole in front after 55 minutes. Van Persie supplied a corner from the right and the Hungarian advanced off his line and tried to punch clear under minimal pressure. However, he only succeeded in turning the ball into the path of Koscielny, who poked the ball over the line.
Arsenal continued to look vulnerable in defence though, with Jenkinson at right-back being given a torrid time by Marc-Antoine Fortune. However, the forward was unable to beat Szczesny after giving his young opponent the slip on numerous occasions.
Van Persie was also required to clear an Olsson header from five yards out, while a flowing move from the home side resulted in substitute Keith Andrews forcing Szczesny into a fine save as he palmed the ball wide of the post.
Arsenal were only threatening in sporadic bursts, with Van Persie blasting a shot off target in injury time, and they almost threw third place away in the dying seconds when Billy Jones burst into the box.
However, Gibbs produced a fantastic saving tackle to ensure that Arsenal will be in the Champions League once again next season.