Lyng added a free for Wexford to settle them, but with Broderick accurate from placed balls and Gry Kelly on the mark, Carlow moved three points clear.
A second yellow card for Eoghan Nolan saw Wexford reducted to 14 men for the last 24 minutes, but they were lifted by PJ Banville’s goal in the 48th minute that left Carlow one point ahead.
Brendan Murphy and Paul Broderick fired Carlow to their first Leinster championship win since 2011 and qualified them for a provincial quarter-final with Dublin at Portlaoise after a 2-17 to 2-13 win over Wexford at Netwatch Cullen Park.
Broderick was immense for the home side and his ten points were crucial, although Brendan Murphy’s two second-half points and his injury-time goal helped his side over the line.
Turlough O' Brien's side trailed by just two points with 62 minutes on the clock and were in with a shot of progression themselves.Four minutes later Wexford’s midfield sector was further weakened when Daithi Waters was black carded for a foot trip on Murphy, while his partner Colm Kehoe was stretchered off eight minutes from half-time with a leg injury.The Carlow forwards took it in turns to run the long role up front, and it was Paul Broderick that showed the best skill and composure in front of goal – he kicked three points in a row from play for his side to draw them level by the 32nd minute.Two minutes later he followed it up with second yellow card though as Carlow’s chief score-getter ended the game on the sidelines, but Brendan Murphy’s goal in the second minute of injury-time wrapped up the win for Carlow.Carlow: C Kearney; C Crowley, S Redmond, C Lawlor; D Moran (1-00), D St Ledger, G Kelly (0-01); B Murphy (1-02), S Murphy; E Ruth, D Foley (c) (0-02), A Kelly (0-01); P Broderick (0-10, 0-05f), C Moran, J Murphy.