Kidwelly then crashed over twice more, through Jason Smith and Rhys Jones, to pick up a 22-9 win. The new Division Two West leaders are Builth Wells, who were also late winners.
They trailed visitors Llandarcy 9-3 going into the last minute before centre John Thurston crossed for a try.
Thurston then put over the resulting conversion, which was the last kick of the game, to see the Bulls nick the game by a point and head the table.
Bridgend Athletic stay in third after a five try victory over Penclawdd. Mark Bennett grabbed a brace of tries, with others coming from Simon Mustoe, Rob Coles and Rod Davies, as Athletic won 38-3.
In the clash of the top two in Division Three West, second-placed Pontyberem reduced the gap on Aberystwyth to just two points after they won 13-5 at the league leaders.
Two unconverted tries from Dan Hawkins and Justin Lloyd, plus a penalty from Mike Sauro saw Bont secure victory at Plascrug.
Division Four West leaders Llandeilo trounced Cwmgors 37-5, and at the top of Division Five West Milford Haven overtook Cefneithin, who did not play, by narrowly beating Aberaeron 19-15.