Albion were 2-0 up just before the hour mark when Peter Odemwingie capitalised on a mix-up between Manuel Almunia and Sebastien Squillaci to add to Steven Reid's early header.
We staged a comeback with goals in the 70th minute from Andrey Arshavin and 78th from Robin van Persie, but we couldn't not find a winner and are now five points behind Barclays Premier League leaders Manchester United, who beat Bolton 1-0.
The Baggies, meanwhile, are a point above the relegation zone in 16th place as they were at the start of the day.
We were in need of a lift following the Carling Cup final defeat and elimination from the Champions League and FA Cup over the last few weeks, but we made the worst possible start as West Brom took the lead after only three minutes.
Albion skipper Chris Brunt swung in a corner and Reid was allowed the space to jump and crash a header beyond Almunia for his first Premier League goal for nearly five years.
Aaron Ramsey - who was starting a game for the first time since breaking his leg last season - appeared to be the main culprit in failing to mark Reid as our defensive frailties were laid bare.
Moments later Jerome Thomas won the ball and broke forward, sending in a dangerous delivery that just evaded Odemwingie.
We started to gain more possession but were struggling to get a proper foothold and the game was temporarily halted after 20 minutes when Laurent Koscielny clashed heads with James Morrison, the latter having to come off for treatment.
While Morrison was off the pitch we showed some threatening signs, with only the head of Jonas Olsson preventing Ramsey getting on the end of a Gael Clichy cross.
We should have then equalised when Van Persie's header came back off the bar.
The ball dropped to Ramsey right in front of goal, but the Wales midfielder could only strike it straight at Scott Carson.
As Morrison returned with a bandage around his head the Gunners continued to press, Samir Nasri seeing a cross come off the legs of Abdoulaye Meite towards goal but Carson was able to avert the danger.
Van Persie tried his luck just before the break but his shot was off-target.
There was a substitution for each side at half-time as Albion boss Roy Hodgson brought on Marek Cech for Paul Scharner, who had also taken a knock to the head, and Wenger introduced Marouane Chamakh in place of Denilson in an attempt to give the side more attacking impetus.
Chamakh soon sent a header goalwards but it did not trouble Carson and within minutes, the Baggies were almost 2-0 in front as Brunt slid in to meet Cech's cross, the captain just poking it the wrong side of the post.
Wenger bolstered his forward line again with Nicklas Bendtner coming on for Ramsey, but just seconds later West Brom did score again - thanks in no small part to Almunia.
The Spanish goalkeeper came charging out of his box when Youssouf Mulumbu launched a long ball forward but completely misjudged it before getting in a tangle with Squillaci and Odemwingie gleefully slotted into the net.
The watching Jens Lehmann, back on the bench after rejoining the club on a rolling monthly contract earlier in the week, looked less than impressed.
When Koscielny picked up a booking for a hefty challenge on Odemwingie and Van Persie blazed a free-kick over the bar, we seemed to be fading, but a slick move saw us suddenly pull a goal back through Arshavin, who played a one-two with Chamakh before turning and rifling a shot beyond Carson.
Marc-Antoine Fortune came on for Odemwingie but we began pouring forward and eight minutes after Arshavin's strike they found a leveller, with the Russian again the catalyst.
His looping cross hit the post and Meite was unable to deal with the rebound before Van Persie bundled it over the line.
We now looked the more likely winners, but Fortune went close at the other end, cutting into the box and seeing his shot deflect wide.
Play switched once more and the Gunners sent a succession of balls into the West Brom box, but there was no way through.
Jack Wilshere shot over the bar and Carson saved from Clichy at the death as Albion hung on for a vital point.