Fourth-placed Phoenix looked to have wasted too many opportunities as the ninth-placed Jets scored twice in the opening 20 minutes and threatened to earn just its second win of the season.
The hard-tackling Jets racked up the fouls, committing 25 to Wellington's 11, and had defender Ben Kantarovski sent off for a second yellow card in the 86th minute.
Both the Jets' goals came from set pieces.
The first came from an 11th-minute Kije Lee corner as Manny Muscat's attempt at a clearing header deflected onto Roly Bonevacia and past goalkeeper Glen Moss for an own goal.
Seven minutes later, Scott Neville met Andrew Hoole's corner at the far post and headed home.
The Jets have been in turmoil recently, sacking skipper Kew Jaliens, David Carney, Adrian Madas, Joel Griffiths and Billy Celeski.
But boosted by the two-goal lead, they dug in and defended grimly, aided by some patchy Phoenix finishing.
The Jets almost added a third when Edson Montano's header ricocheted off the post after quick work down the left from Lee. Sam Gallagher's nicely timed cross stretched Phoenix on the counter-attack.
Little changed after the restart, with Phoenix continuing to create chances but unable to find the killer finish - until the 68th minute.
Quick work from Ben Sigmund and Nathan Burns set up Roy Krishna to toe the ball home at the far post.
Newcastle's scrambling defence then held until Hicks curled the late free kick high into the right hand corner.
Wellington coach Ernie Merrick was happy enough with the result and the wealth of chances his team created.
"I thought it was a tremendous display of football without the finishing," he said.
"We conceded two soft corner kicks, but apart from that we never really looked under pressure. We had more than enough chances to win the game.
"The boys are really dejected but I'm not. Just because that final touch doesn't go in the goal, that doesn't mean we didn't play some terrific football."
Jets coach Phil Stubbins was well pleased with the way his under-pressure team backed up from last week's 1-1 draw with Western Sydney Wanderers.
"We can come away from the last three games with our heads held high, we've had three very close games now and we've certainly worked hard enough for the win," he said.
"The kids that are out there will be learning a hell of a lot. I think we've taken another step forward in terms of education today."
With just one win from 18 starts, Newcastle is well out of finals contention, but Stubbins says there is still plenty of motivation for his young squad.
"We're just focusing on every game, one game at a time, and we did ourselves justice today. We'll be trying to build on what we've done."
