West Brom fought back strongly to take a 2-1 first-half lead at Upton Park, Nicolas Anelka scoring a brace after Joe Cole had put West Ham in front after four minutes.
Modibo Maiga and Kevin Nolan put West Ham ahead for a second time, only for Saido Berahino to seal a point with a superb individual strike after 69 minutes.
West Brom came from a goal down to earn draws against both Hull City and Tottenham in their two previous encounters, and Downing hailed his players for displaying great spirit once again.
"That (the character) comes from the players," Downing said. "They take a lot of credit from the fact that they grow when we concede.
"From Hull onwards, through Tottenham and today - twice coming back - I think that goes down to the players' attitude to get themselves back into the game."
Downing has now been in charge for three games following Steve Clarke's sacking earlier this month.
Sporting and technical director Richard Garlick insisted the club are making progress in their search for a new boss on Friday, but Downing has not received any update regarding a potential appointment.
"All I've been told is to carry on," Downing added.
"I will do that again for the new year, and if anything comes before that I will gladly step down and support the guy that comes in."
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