But that second-minute score was soon pegged back as Wasps' full-back Mark Van Gisbergen kicked the first of his two penalties and then the side went ahead with a try for hooker Ben Gotting, only for the visitors to see rookie flanker Alan Wyn-Jones reclaim the lead for the home side with an 18th-minute score.
But three minutes later Van Gisbergen put his side ahead with his second penalty and after 35 minutes England centre Stuart Abbott crossed for the full-back to convert.
And despite the Ospreys' best efforts London Wasps held out for the victory.
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