Ulster responded well after the restart and Herring, perhaps lucky to be on the pitch after his earlier yellow was not referred to the bunker for review, crossed for his second try of the game.
Doak converted, but just like in the first half, the Dragons had an immediate response as Reed landed his second penalty of the match.
A big turning point came on the hour mark, when Ulster pressed the Dragons line and looked to have burrowed over for a try but some superb Dragons defence won the penalty turnover.
That moment was only compounded when Reed kicked the Dragons nine points clear with a penalty, before replacement Cooney swiftly responded with a kick of his own.
Reed had another opportunity to restore the nine-point margin but his kick came back off the post.
That miss was punished as Wilson stole the ball at a ruck in the Dragons’ 22, spun away from a tackle and showed tremendous power to cross, and Cooney converted to give Ulster the lead for the first time.
John Andrews thought he had added a fifth try in the final minute but it was ruled out for obstruction, but play was brought back for a penalty and Cooney landed a simple kick to help Ulster over the line.
Dragons: Huw Anderson, Rio Dyer, Joe Westwood, Aneurin Owen (capt), Ashton Hewitt, Will Reed, Rhodri Williams; Rodrigo Martinez, Elliot Dee, Chris Coleman, Matthew Screech, George Nott, Shane Lewis-Hughes, Harrison Keddie, Aaron Wainwright.
Replacements: Brodie Coghlan, Dylan Kelleher-Griffiths, Paula Latu, Steve Cummins, Taine Basham, Dane Blacker, Josh Thomas, Jared Rosser.
Ulster: Stewart Moore; Michael Lowry, James Hume, Stuart McCloskey, Jacob Stockdale; Jack Murphy, Nathan Doak; Callum Reid, Rob Herring, Tom O’Toole; Alan O’Connor (capt), Matthew Dalton; David McCann, Nick Timoney, James McNabney.
Replacements: Andrew, Crean, Wilson, Treadwell, Crothers, Cooney, Morgan, Ward.
Referee: Griffin Colby (SARU)
Assistant referees: Rhys Jones (WRU), Gwyn Morris (WRU)
TMO: Morne Ferreira (SARU)
