Hodgson, sacked by Liverpool last month, has replaced Italian coach Roberto di Matteo after the Italian coach was relieved of his duties following a run of 13 defeats in 18 matches in all competitions.
Hodgson was at The Hawthorns on Saturday to watch his new charges throw away a 3-0 half-time lead to draw 3-3 with West Ham United, leaving the Baggies above the relegation zone on goal difference.
Asked if West Brom had the players to avoid the drop, Hodgson replied: "I'm afraid after 90 minutes working with them today (Monday) and 90 minutes watching them on Saturday, I can't answer that question.
"I would like to say so, but it would be foolish of me to make rash statements."
Dismissed by Liverpool in January after a 3-1 defeat at Blackburn left the 18-time champions in 12th place, Hodgson admitted he had not expected to return to work so swiftly.
"I wasn't actually planning to come back to the Premier League and keep working so soon after leaving my last job at Liverpool, but (technical director) Dan (Ashworth) and (chairman) Jeremy Peace were very persuasive in the picture they painted of the football club," said the 63-year-old.
"We don't have much time. Like a lot of other clubs, we're in a desperately difficult situation at the bottom of the league, trying to retain our status.
"That is the major goal and ambition, and if we achieve that we want to build a strong platform for next season."
The relegation dog-fight is a far cry from the lofty objectives Hodgson was pursuing when the season began at Liverpool but he insisted that the pressures at The Hawthorns would be just the same.
"All jobs are different, there is no question of that," said Hodgson, who led Fulham to the final of the Europa League in 2010 before joining Liverpool.
"The Premier League these days is a tough league, a very interesting league for any club that's in it, whether you're punting for a place in the top four or a club that would be very happy just to still be in it.
"Jobs in this league are precious."
West Brom host bottom club Wolverhampton Wanderers in their next league game on Sunday.
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