Striker Wood, however, popped up in the 85th minute to poke the ball home from close range in a congested box after Honduras goalkeeper Luis Lopez had done well to parry a free kick.
The result left the U.S. and Honduras both on nine points after eight matches in the CONCACAF qualifying group, known as the Hexagonal.
Both teams fell behind Panama, who moved into third place after beating Trinidad and Tobago 3-0 later on Tuesday night.
Second-placed Costa Rica could have guaranteed their place in Russia but they only managed a 1-1 draw at home to Mexico, the group leaders who cemented their place last week.
Mexico got the opener after 42 minutes thanks to an own goal from Cristian Gamboa but the home side grabbed a stunning equaliser seven minutes from time when Marco Urena volleyed home from 20 yards out.
The top three teams in the group will qualify for Russia with the fourth-placed nation going into a playoff against either Australia or Syria.
"We have a lot of work to do but it keeps it in our hands now," U.S. striker Jordan Morris said after the game.
"Conditions were difficult. We knew we had to get a result down here. We worked really hard the whole game.
"Would have loved three points but a point is big for us down here. I love playing with Bobby. He's a good finisher."
The U.S. have played in every World Cup since 1990.
American coach Bruce Arena had made seven changes to the line-up that started the 2-0 home loss against Costa Rica last Friday.
Honduras dominated the first half and took a well-deserved lead in the 27th minute when Quioto seized on a mistake by American central defender Omar Gonzalez and curled a right-foot shot past goalkeeper Brad Guzan.
Woods came on with less than 20 minutes remaining and made no mistake with the one good chance that came his way.
(Reporting by Andrew Both in Cary, North Carolina, editing by Nick Mulvenney)
