Almagro, 10th in the world rankings, needs a strong finish to the season to move into the top eight and secure a place at the 02 Arena in London.
But the ninth-seeded Spaniard was made to pay for taking just one of eight break-point opportunities en route to a 4-6 7-6 (7/4) 6-2 defeat by former world number two Haas, who will face Tommy Robredo next after the Spaniard beat Alex Bogomolov Jr 7-6 (7/4) 6-4.
It was a better day for fellow seeds Marin Cilic and Richard Gasquet, both of whom booked their places in the next round with straight-sets victories.
Tenth seed Cilic had suffered first-round exits in the last three years in China but had few worries in seeing off Polish qualifier Lukasz Kubot 7-5 6-3 while Gasquet, seeded 11th, swept past Brian Baker 6-2 6-3 in 70 minutes.
Grigor Dimitrov set up a second-round clash with Novak Djokovic after a 7-5 6-3 win over Pablo Andujar, Sam Querrey routed Zhe Li 6-3 6-1, firing down 11 aces in the process, and Benoit Paire battled past German qualifier Philipp Petzschner 7-6 (7/2) 6-4.
Kevin Anderson progressed after a 6-2 7-6 (7/4) victory over Jarkko Nieminen, Andreas Seppi eased past Michael Berrer 6-2 7-6 (7/5) and Alexandr Dolgopolov dismissed Jeremy Chardy 6-3 6-2.