From the restart, Murphy kicked back downfield but Toby Faletau gathered and ran it back, a move that ended in Wardle's try.
Tom Prydie added a further penalty to leave Munster requiring a converted try to win and though they tried valiantly the home defence held firm.
Munster trailed 19-5 at the break and then went further behind when Prydie landed a 48th minute penalty.
To their credit they came roaring back and tries from Ronan O'Mahony - who made an impressive debut - replacement CJ Stander and then Murphy saw them edge in front before Wardle and Prydie denied them.
Munster:Denis Hurley; Ronan O'Mahony, Ivan Dineen, JJ Hanrahan, Johne Murphy; Ian Keatley, Cathal Sheridan; Wian du Preez, Damien Varley, Stephen Archer; Dave Foley, Billy Holland; Dave O'Callaghan, Niall Ronan, Paddy Butler. Replacements: Sean Henry, Marcus Horan, John Ryan, Ian Nagle, CJ Stander, Duncan Williams, Cian Bohane, Danny Barnes.
Dragons: Tom Prydie, Matthew Pewtner, Ross Wardle, Lewis Robling, Tonderai Chavhanga, Steffan Jones; Jon Evans; Aaron Coundley, Hugh Gustafson, Nathan Buck, Andrew Coombs capt, Rob Sidoli, Dan Lydiate, Nic Cudd, Toby Faletau Replacements: Sam Parry, Owen Evans, Tim Ryan, Adam Jones, Jevon Groves, Ieuan Jones, Wayne Evans, Andy Tuilagi