The U S Open champion, needing a win after losing to Britain's Andy Murray when round-robin action got under way on Sunday, failed to take a match point at 5-3 in the decider and saw another slip away in the following game as Verdasco hung on.
He forged 6-1 ahead in the tiebreak, however, and this time made no mistake, securing victory when Verdasco hit a forehand into the tramlines.
Verdasco must win his final match against Murray on Thursday, and hope other results go his way to have any chance of reaching Saturday's semi-finals. Del Potro will next face Roger Federer in his final match on Thursday.
The lanky Argentine, who is ranked fifth in the world rankings, showed the form that so impressed in New York as he took control of the first set.
But Del Potro, who has been below his best since winning his maiden Grand Slam title at the US Open in September, toiled in the second set and a wayward game handed Verdasco the second set.
The final set saw a see-saw battle between both the players, both who were fighting to stay in the competition. Del Potro broke in the sixth game when he successfully challenged that Verdasco's forehand was out.
But Verdasco fought his way through from being 2-5 down to take it to the tie-break but was unable to deliver when it mattered the most, handing Del Potro the win in the two hours and 23 minutes battle.
Tournament debutant Verdasco is now relying on Roger Federer beating Murray in the evening match in Group A to keep alive his slim hopes of reaching the semi-final.
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