Sam Allardyce's side headed to Gelsenkirchen to take part in the Schalke Cup along with the hosts, Newcastle United and Malaga, and they emerged with two points after Jaaskelainen saved two spot-kicks in the shootout.
West Ham saw plenty of the ball throughout the encounter, but rarely troubled Fabian Giefer in the Schalke goal as their final ball was often found lacking.
Schalke, though, carried a very potent threat and Christian Fuchs almost gave them the lead in the 25th minute, only to see his 25-yard effort fly just over.
Allardyce's men had an even narrower escape 10 minutes later as Eric Choupo-Moting charged down the left flank, cut inside James Collins and then struck the left-hand post.
Although Schalke looked the more menacing, West Ham continued to enjoy their fair share of possession going into the second half and Collins headed wide from a corner.
Schalke almost made them pay for that miss seven minutes later as Marvin Friedrich pounced on a loose ball in the West Ham area, but the centre-back's effort came crashing back off the crossbar.
Neither side could find the net in open play, forcing the match to be decided on penalties and, although Ricardo Vaz Te's off-target kick gave West Ham an uphill struggle, Jaaskelainen denied both Felipe Santana and Axel Borgmann to claim victory.
