© Getty Images
The result at the Hunter Stadium leaves Newcastle needing Perth Glory to beat Sydney FC by at least two goals to see them qualify for the A-League finals.
After a competitive first 20 minutes, the Jets took the lead when Griffiths got on the end of a corner from David Carney to powerfully head into the net.
Adelaide began the second half on the front foot, but Newcastle gradually eased themselves back into the action, with Griffiths being denied by a fine save by Eugene Galekovic.
Newcastle stopper Ben Kennedy kept his side in the game as the match neared a conclusion, and his interventions proved crucial as the Jets sealed the win in the final minute when Taggart netted from close range after being set up by Michael Bridges.