Premier League: Southampton 0 West Ham 0
The home side created the better chances throughout the 90 minutes, but were unable to find a way past the impressive 38-year-old on Sunday.
Pablo Daniel Osvaldo was the first to be frustrated by the Finn in the first half when the veteran goalkeeper denied him with a point-blank save.
Southampton continued to look the more likely to break the deadlock after the break with Rickie Lambert hitting the post, while Jaaskelainen reacting well to a half-volley from Morgan Schneiderlin.
Mauricio Pochettino handed starts to Nathaniel Clyne and Jay Rodriguez, while Lambert made his 200th appearance for Southampton.
Ravel Morrison made his first Premier League start for Sam Allardyce's side at the expense of the injured Stewart Downing (leg).
After a tepid opening it was West Ham who came closest to making the breakthrough in the seventh minute.
Mohamed Diame collected the ball on the right-hand side of the box and bustled past Luke Shaw before forcing Artur Boruc to parry away his drilled effort at the near post.
Osvaldo was then denied his first Premier League goal when he drew a superb point-blank save from Jaaskelainen after good work from Rodriguez to centre from the left.
With both teams struggling to create clear-cut chances, Lambert will feel he could have done better in the 23rd minute when his attempted pass to Osvaldo deflected back into his path on the edge of the area. However, his first-time volley was tame and easily held by Jaaskelainen.
Ten minutes later, Schneiderlin irked the West Ham players with a late lunge on Diame, the defender's challenge was mistimed and he was perhaps lucky to escape punishment.
Morrison nearly marked his first Premier League start with a goal five minutes before half-time, but his 20-yard effort curled wide.
Southampton nearly made the ideal start to the second half in the 48th minute when Victor Wanyama's dipping half-volley was tipped over the crossbar.
Lambert came closest to opening the scoring from the resulting corner when he headed Adam Lallana's delivery against the post.
The hosts continued to look the more likely to break the deadlock, and Schneiderlin's close-range effort rebounded into the air where Osvaldo tried an audacious overhead kick that was headed away by Winston Reid.
Jaaskelainen once more came to West Ham's rescue in the 67th minute. The lively Rodriguez crossed from the right and Schneiderlin's half-volley forced the goalkeeper into a superb reaction save to his left.
Lambert was the next to be thwarted by the West Ham stopper.
Rodriguez sprinted through the middle and laid into the path of the England international, who opened up to a curl his effort towards the far corner where Jaaskelainen produced another tremendous save.
West Ham almost made Southampton pay for their profligacy when a scramble in the area saw the ball fall to the unmarked James Collins but the centre-back blazed over the bar from eight yards.