Fulop, who replaced the injured Ben Foster in the Albion goal, was at fault for all three Gunners goals from Yossi Benayoun, Andre Santos and Laurent Koscielny at The Hawthorns as Arsene Wenger's side edged out Tottenham for third spot.
Albion had netted through Shane Long and Graham Dorrans after Benayoun's opener, but Fulop's horror show meant it was a losing end to new England head coach Roy Hodgson's reign.
Wenger will take immense satisfaction from his side's achievement after some supporters had called for him to be sacked earlier in the campaign.
It will also give him a stronger hand to play in attempting to keep hold of leading scorer and skipper Robin van Persie.
Albion will now look to find a successor for Hodgson and attempt to build on a second successive mid-table finish.
Arsenal were the first to settle and a mistake by Fulop gifted Benayoun the opening goal after three minutes.
Van Persie and James Morrison challenged for the ball in midfield and the ball ricocheted into the path of Fulop.
But he appeared to stumble when under no pressure and Benayoun was left with the simplest of tasks to knock the ball into an empty net.
Albion retaliated and Wojciech Szczesny produced a fine low save to keep out a shot on the turn from a narrow angle by Marc-Antoine Fortune.
But after 11 minutes the fit-again Long brought the Baggies on level terms.
He chased down a pass from Morrison and hammered a low shot past Szczesny into the corner of the net.
Wenger appealed in vain for offside with the fourth official but Long appeared to be played onside by Carl Jenkinson.
Three minutes later an equally clinical finish from Dorrans put the Baggies in front.
Morrison was again the creator and Dorrans gave Szczesny no chance of saving with a similar finish to that of Long.
Gareth McAuley blocked an effort from Benayoun inside the box and a Van Persie free-kick hit the defensive wall, but after half an hour Santos brought the Gunners level.
Fulop got a hand to his swerving left-footed drive from 20 yards out but could only parry it into the corner of the net.
Play was moving from end to end and Van Persie became the first player to be booked seven minutes before half-time for a touchline challenge on McAuley.
Wenger brought on Theo Walcott in place of Tomas Rosicky for the second half.
The visitors started the period on the offensive and a cross-shot from Santos was only just off target.
Albion were forced to reshuffle after 51 minutes when the impressive Morrison limped out of the action to be replaced by Keith Andrews.
Fulop's horror show continued when another mistake gifted Koscielny a goal after 54 minutes to restore Arsenal's lead.
The former Sunderland goalkeeper punched a corner from Van Persie back towards his own goal and Koscielny made no mistake from four yards out.
Albion tried to recover and a Jonas Olsson header was deflected over his own bar by Van Persie.
Then Szczesny reacted quickly to turn aside a shot on the turn from Andrews.
In injury-time substitute Kieran Gibbs produced a vital block on Billy Jones' shot after he burst into the box.