A win in their final match against Trinidad and Tobago will likely be enough to see the United States through as Honduras would need to win both their remaining games and overturn a goal difference currently at 12.
Top team Mexico are already through from the North, Central American and Caribbean Confederation and second-placed Costa Rica need just a point from their final two matches to join them.
Mexico came from behind to beat the already eliminated Trinidad and Tobago 3-1 at home on Friday.
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Bottom side Trinidad had won just one of their previous eight matches but Shahdon Winchester gave them a surprise lead after 66 minutes.
However, Hirving Lozano got the equaliser after 78 minutes and Javier Hernandez popped up to save Mexico’s blushes two minutes from time. Hector Herrera capped the scoring with a free kick in injury time.
Costa Rica's scheduled qualifier at home to Honduras was postponed until Saturday from Friday due to severe rainfall caused by tropical storm Nate.
The top three teams qualify automatically, while the fourth-placed side goes into a play-off against a team from the Asian confederation.
Pulisic got the United States' opener after eight minutes when he sprinted free of the defence and slotted the ball home from a tight angle after rounding the goalkeeper.
Altidore made it two 11 minutes later when he side-footed home an inviting low cross from Pulisic. The burly striker made it 3-0 just seconds before half time when he chipped home a penalty after Bobby Wood was upended in the box.
Wood got the goal his play deserved 18 minutes into the second half when he spun inside the box and fired past goalkeeper Jaime Penedo.
The United States should have scored more but profligate finishing and a fine performance from Penedo kept the score down.
"On the night we actually didn’t finish well," said U.S. coach Bruce Arena. "We could have scored a lot more goals.
"Now we have to finish it off Tuesday in Trinidad."
(Reporting by Andrew Downie; Editing by Peter Rutherford)
