The Gunners produced a fairly listless performance at The Hawthorns in the absence of the injured Mesut Ozil, but Alexis Sanchez's deflected free-kick seven minutes from time looked to be enough for them to snatch all three points.
However, a controversial late penalty allowed Jay Rodriguez to score the equaliser and lift West Brom off the bottom of the table on goal difference, while Arsenal move into fifth above arch-rivals Tottenham, but are three points off the Champions League spots.
Ozil was left out of the squad due to a knee injury that could also see him miss Wednesday's clash with Chelsea, and Wenger was given a further headache when Sead Kolasinac limped off during a tepid first half.
Sanchez, who continues to be linked with a move to Manchester City, started alongside Alexandre Lacazette and Alex Iwobi in attack, but clear-cut opportunities were scarce against Alan Pardew's disciplined side.
The visitors' pressure increased in the closing stages and Sanchez struck the vital blow when his free-kick glanced in off James McClean, but Rodriguez levelled in the 89th minute to leave Wenger furious.
Arsenal were dealt a blow before kick-off, with Ozil missing from the squad due to a knee injury, and goalkeeper Petr Cech then picked up a knock after dropping a cross under pressure in the early exchanges.
Cech was unsettled for some minutes but he did manage to claw away a header from Rodriguez, while Arsenal's best early chance saw Iwobi curl a shot over the crossbar from the edge of the box.
Iwobi was then denied by Ben Foster after creating some space 20 yards out, but the Gunners' drab half got worse nine minutes before the break when Kolasinac limped off after a foul by Gareth Barry.
West Brom's pressure began to increase after half-time, but Foster had to be alert to block Lacazette's low strike after the France international made a rare drive into the penalty area.
Arsenal suddenly looked a little livelier in attack and Lacazette prodded a promising chance just wide of the left-hand post, before Sanchez rippled the side-netting with a dipping effort from the right.
Wenger introduced Per Mertesacker for Koscielny, who appeared to be moving a little gingerly, before throwing on Danny Welbeck in a bid to break down a West Brom defence that was beginning to drop deeper and deeper as the clock ticked down.
Sanchez thought he had won the match for Arsenal with seven minutes left, his free-kick glancing off McClean in the wall and nestling into the bottom corner in front of the visiting fans.
However, with a minute of normal time remaining, referee Mike Dean penalised Calum Chambers when Kieran Gibbs' cross struck his hands from close range, allowing Rodriguez the chance to fire the ball down the middle of Cech's goal and rescue a point.
Key Opta stats:
- Arsenal have scored in all 24 of their Premier League games against West Brom – the best 100% record in the competition.
- Petr Cech has failed to save any of the 13 penalties he's faced in all competitions for Arsenal – only two keepers have faced more at a single club without saving one (Brad Friedel at Aston Villa, Craig Gordon at Sunderland both 15).
- The Baggies are now winless in 19 Premier League games – only three clubs have ever had a longer winless run in the competition (Derby 32, Sunderland 26 and Norwich 21).
- This was the first point that West Brom had won from a losing position in the Premier League this season.