Opener Jordan Smith struck a useful 47 before being trapped in front by Habib Naqvi, but the majority of the batting lineup struggled, with the Bath-based side eventually bowled out for 170.
Knowle set about trying to achieve the score with wicketkeeper-batsman Matthew Wilson striking a well-crafted 57 from 104 balls, but Chris Day had him caught by Josh Smith.
Two wickets for Thomas Hankins stemmed the Knowle tide, but Ian Mills' unbeaten 53 saw the visitors home with six overs and two balls to spare meaning Lansdown picked up just the four bonus points.