The Exiles remain three points outside the top seven but, with one game left, only a mathematical miracle would now allow them to climb above Plymouth, who have a superior goal difference of 18.
Dack’s team ended a run of four straight defeats with second-half goals from David Tutonda and Lee Minshull.
In between, Jake Hyde had a home penalty saved by County keeper Joe Day after Michael Coulson had also shot over an open goal.
On a day of mixed emotions, Dack said: “I thought we were outstanding from start to finish and I thought we were worthy winners.
“All in all, it was a great day’s work and we look forward to our last game now. We showed our supporters what we’re all about as a group of players.
“We’ve never been involved in a promotion bid at this level. There’s a lot of pressure in these games and the players will be better for it next season. As a club, we have massively over-achieved.
“When you look at other clubs compared to us, we don’t even own our own ground or a training facility. We’ve pushed the boundaries to the limit.
“Everyone is working really hard to get this club in the play-offs. For us to have got there would have been an unbelievable achievement but our goal difference was always going to be bite us.”
Minstermen boss Russ Wilcox, meanwhile, bemoaned his team’s flat finish to their home fixtures but also stressed that the biggest priority had been to keep the team in the Football League on his arrival in mid-October.
“If we look at the small picture, then this game was a disappointment but, if we look at the bigger picture, we could be sat in Cheltenham or Tranmere’s dressing rooms now,” he reasoned.
“Our performance wasn’t good enough on this occasion and there were pivotal moments when Michael Coulson missed his chance and Jake Hyde had the penalty saved.
“They were the better team and I thought their two wing-backs caused us problems, so I’m disappointed we didn’t finish the season on a high with four successive home wins. But the main thing is we will playing League Two football next season.
“It’s job done in that respect and we can look to perform surgery now over the summer to try and make sure we are competing at the other end of the table next season.”
Tutonda opened the scoring from 10 yards on 50 minutes after Yan Klukowski had pulled the ball back from the byline.
After good work by on-loan Spurs striker Shaq Coulthirst, Coulson then lifted a simple chance over Day’s vacated goal.
Day went on to save Hyde’s 62nd-minute penalty low to his left before Minshull made the points safe with a free header from Mark Byrne’s left-wing free kick on 75 minutes.