Pugh grabbed the winner in Saturday’s League One clash at Dean Court when his whipped-in cross nestled in the back of the net in the 30th minute as Rovers goalkeeper Paul Rachubka misjudged the flight of the ball in a gusting wind.
Tranmere had taken a fourth-minute lead through winger Martin Devaney, who latched on to a clearance from Rachubka to fire home from just inside the box in the fourth minute.
Bournemouth pulled levelled 14 minutes later as Wes Thomas fortuitously grabbed his 10th league goal of the season.
Swiveling on the edge of the box, the striker’s shot took a massive deflection off Mark McChrystal and looped into the far corner of the visitors’ goal.
The second period turned into a rather drab affair with both defences keeping the attackers at bay.
Tranmere substitute Lucas Akins produced the most threatening moment only to see his stinging drive well held by home goalkeeper Darryl Flahavan.