West Brom (18 points) are fourth from bottom, one point ahead of Crystal Palace and Burnley with Leicester City in 20th and last place on 14.
The coming weeks, though, could see several other clubs drawn into the annual scrap against the drop with Queens Park Rangers and Hull City one point in front of Albion.
Former Palace and Stoke City boss Pulis has never been relegated as a manager but said that counted for little.
"Irrespective of my record...we're out in the open and everyone is fighting to stay up," he told reporters.
"Just because I've walked through the door it doesn't mean we're going to stay up. There's got to be a lot of work."
Sunderland have 20 points while Everton (21) are in freefall after four straight defeats.
Aston Villa (22) have pulled clear of the three relegation places but after being too close for comfort in the past three seasons they know they can take nothing for granted.
Nine clubs can feel as though they are in danger and what makes this season's battle particularly intense is that no team is cast adrift at the bottom.
Results over the festive period showed there was plenty of fight among the strugglers, Burnley showing the way by coming from behind to get draws at champions Manchester City (2-2) and Newcastle United (3-3).
Leicester rallied from two goals down to draw 2-2 at Liverpool on Thursday after having won 1-0 at Hull.
Steve Bruce's Hull bounced back on New Year's Day with a 2-0 victory over Everton.
West Brom's perilous position led to manager Alan Irvine being sacked last week and Pulis taking over.
Palace also made a managerial change last week, sacking Neil Warnock with Alan Pardew expected to join them from Newcastle.
The January transfer window adds another element of excitement. There is likely to be plenty of activity from clubs at the bottom but finding the forwards to get the goals that will keep them up will be easier said than done.
"Everyone is looking for strikers so it will be tough in this window," said QPR boss Harry Redknapp.
(Reporting by Simon Evans, editing by Tony Jimenez)
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