Pulis was appointed head coach at The Hawthorns earlier in the day - replacing Alan Irvine - and watched from the stands as Saido Berahino cancelled out Diafra Sakho's opener before half-time.
His strike ended a run of three straight Premier League defeats for the Midlands club while Sam Allardyce's West Ham failed to get back to winning ways after ending 2014 with back-to-back defeats to against Chelsea and Arsenal.
Despite West Brom starting well, Pulis watched on as his backline was carved open by a clever ball from Morgan Amalfitano, Aaron Cresswell subsequently crossing for Sakho to head home early on.
The good and bad sides to West Brom were then on show for Pulis as the visitors wasted several chances before the recalled Berahino eventually equalised three minutes from half-time.
Allardyce's side were much improved after the break but neither team was able to find a winner as West Ham remain sixth while West Brom stay a point clear of the drop zone in 17th.
Eager to impress their new boss, West Brom began brightly with Stephane Sessegnon and Berahino helping the visitors start on the front foot.
West Ham's first foray forward came about from a slick passing move nine minutes in, which saw Cresswell stumble inside the area before he could connect with a shot on goal.
However, Cresswell was then instrumental in giving West Ham their lead when crossing for Sakho to head home his eighth Premier League goal 10 minutes in.
Former West Brom man Amalfitano played Cresswell in behind Andre Wisdom and the full-back's delivery was met by Sakho, whose header found its way past Ben Foster off the far post.
Stewart Downing tested Foster from close range soon after, with opposite number Adrian saving well from Chris Baird's volley as West Brom forged their first opening.
As in the Boxing Day defeat to Manchester City, West Brom continued to waste chances as Sessegnon blazed wide when afforded time and space in the area.
Allardyce had brought Carl Jenkinson and James Collins into his defence and they were tested further when Chris Brunt evaded Collins - only to divert Sessegnon's pass wide before Berahino finally levelled shortly before the break.
Sessegnon - who shook off a knock to keep his place in caretaker Rob Kelly's starting XI - was again influential, bursting forward before squaring for the striker to sidefoot home.
Berahino was denied a second early in the second half by a sprawling Adrian as Kelly's men continued to impress before the home side eventually managed to regain a foothold in the game.
Cresswell's free-kick struck Foster's left-hand post while Downing's cross from the right led to a goalmouth scramble, but the hosts failed to regain the lead as the spoils were shared in east London.
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