The loss puts Cardiff in grave danger of relegation from the Premier Division, joining their regional Blues side, who are warm favourites to lose their Heineken Cup spot next season.
Promising Cardiff fly-half Nick MacLeod also scored two tries in an entertaining match, but 18 points from the boot of Andrew James ensured Bedwas returned home with the points.
With two matches left, Cardiff sit on 27 points in the table with Llandovery on 26. Bottom-placed Llanelli are also on 26, but have an extra game to play.
Llanelli kept alive their hopes of survival with a 32-23 victory at home to Cross Keys, with ex-Wales winger Richard Rees snaring two tries, while Llandovery suffered a heart-breaking last-minute 28-30 loss to Ebbw Vale.
In other games, Carmarthen Quins fly-half Giles Thomas celebrated his 150th League match by scoring all the points in his side's 19-10 win over Pontypool, while Newbridge out-scored Swansea six tries to four, but lost 34-43 thanks to the boot of All Whites fly-half Luke Richards (23 points).
Results: Cardiff 33 Bedwas 22, Carmarthen Quins 19 Pontypool 10, Llandovery 28 Ebbw Vale 30, Llanelli 32 Cross Keys 23, Swansea 43 Newbridge 34.