Modibo Maiga had two early chances for the home side, but the striker lifted both of his efforts over the bar.
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At the other end, a Matt Jarvis cross broke loose in the Sunderland penalty area, but a number of Black Cat defenders put their bodies on the line to deflect James Collins's shot over the bar from six yards.
Fabio Borini fired straight at Jaaskelainen with Altidore in a good position inside the penalty area and Phil Bardsley rattled the crossbar with a 25-yard shot, as the visitors looked more likely to break the deadlock.
With the game approaching half time, Guy Demel and then Jaaskelainen thwarted Emanuele Giaccherini and Lee Cattermole respectively as the home side struggled to the break with their clean sheet intact.
Sunderland continued to create chances in the second period and a good move involving Borini and Ki Sung-Yeung saw Altidore presented with a good opportunity, but the striker's effort deflected behind for a corner off Demel.
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Jaaskelainen continued to frustrate Altidore, who saw his shot pushed wide by the Finnish keeper on the hour mark.
Both sides made attacking changes as the game approached the final 20 minutes, with Adam Johnson and Steven Fletcher entering the fray for the visitors, while Carlton Cole came on for West Ham, and Mohamed Diame flashed a volley wide of the post in the 75th minute.
Jaaskelainen was on hand once more to tip Ki's curling effort away 10 minutes from time, but it was West Ham who could have nicked all three points in injury time.
Collins rose highest in the area to meet Mark Noble's corner, but the Welshman nodded over from nine yards out.