The visitors were the better side for much of the game, but had to settle for a point after County's Alan White struck in the 67th minute to cancel out Fola Onibuje's opener.
Wise was convinced Lee Peacock had scored a winner after his shot bounced down off the crossbar, but the referee ruled otherwise and had to endure some angry words from the ex-England midfielder.
For their part, Notts stretched their unbeaten home run to five but never got on top of Swindon in the way they had dominated Middlesbrough in winning the midweek Carling Cup tie.
Swindon settled better at the start and it was not long before Notts goalkeeper Kevin Pilkington was in action, tipping an Onibuje effort over the crossbar.
Moments later Swindon's midfield maestro Paul Evans tested Pilkington and then, five minutes before half-time, came the biggest let-off of all for Notts, as a crashing effort by skipper Ady Williams came back off the bar with Pilkington for once beaten.
Notts knew they had to improve in the second half but within two minutes of the restart they were behind. White and Pilkington got in a terrible mix-up and Onibuje lobbed the ball into the far corner of the empty net.
Within minutes Pilkington had to make a smart save on his six-yard line from the same player.
Notts raised their game and Jerel Ifil saved Swindon with a crucial block to deny Andy Parkinson before Notts hauled themselves level on 67 minutes.
As so often this season, it was a set-piece which produced the goal. Ian Ross' corner was headed through by White for his third goal of the season as he atoned for his earlier error.
Swindon were the more dangerous in the closing stages and thought they'd regained the lead after Peacock's rasping drive cannoned down off the bar in the 78th minute.
A minute later Pilkington denied Swindon substitute Christian Roberts by pushing his shot away to concede a corner, and it continued to be keenly fought right to the end.