His first try came with only three minutes on the clock with Powell's conversion followed by a Nicky Thomas penalty.
Bedwas should have increased their lead but sloppy handling saw them fail to capitalise on good opportunities. However, with Powell returning to form with some excellent place kicking, Bedwas extended their lead to 13-3 before Swansea winger Richard Williams collecting an awkward bounce from the restart and quickly moved the ball left for centre Alec Jenkins to cross for an unconverted try.
Powell and Thomas exchanged penalties as half time approached with Bedwas 16-11 up at the break.
The penalties continued in the second 40 before Cameron Clement saw yellow for the visitors.
Bedwas sent on several replacements shortly afterwards and they paid immediate dividends, Budina adding two quick scores to complete his treble. Powell converted to bring up his 21-point haul.
The home side refused to take the win for granted though and veteran Deri Mattravers' try secured a bonus point.
Swansea had the last word deep into injury time when prop Tom Sloane crossed out wide but there was no doubting the overall winners of the contest.