Turnbull earned Motherwell a 4-3 win when he lashed home with his left foot after his right-foot strike had bounced off the bar following a low cross from fellow teenager Jamie Semple.
The late blow left Dundee seven points adrift with three games left and staring relegation from the Ladbrokes Premiership in the face.
Jim McIntyre's side – who had only scored once in their previous seven matches – twice took the lead inside the first quarter through Martin Woods and Ethan Robson, but Motherwell hit back instantly each time through Turnbull and James Scott.
Nathan Ralph equalised swiftly after Gboly Ariyibi put Motherwell ahead early in the second half and Dundee got a lifeline when Alex Gorrin was sent off for the hosts in the 58th minute.
But Motherwell refused to hold on to what they had and Turnbull netted his second deep into five minutes of time added on.
The size of Dundee's travelling support said everything about the belief among their fans in their chances of survival. After more than 2,000 travelled to Perth last weekend, only 302 made the journey to Lanarkshire.
The few remaining optimists were given cause for hope when their team won a penalty as a grounded Ariyibi brought down Cammy Kerr after the Dundee right-back emerged with the ball following a sliding tackle.
Woods gave the visitors an 11th-minute lead from the spot but Turnbull's deflected strike from long range looped over Seny Dieng and into the net 60 seconds later.
Dundee regained the lead in the 22nd minute when Robson held off a defender and curled into the far corner from 20 yards. But Scott grabbed his first Motherwell goal two minutes later when his shot squeezed through Dieng and inside the far post following a corner.
The chances continued at both ends without either goalkeeper being called into serious action again before half-time and Richard Tait thrilled the home fans with a Cruyff-turn nutmeg on Jesse Curran.
The second half was equally open. Mark Gillespie diverted Jesse Curran's volley on to the post before Ariyibi fired Motherwell ahead in the 50th minute after persistent play from Scott.
This time it was Dundee's turn to level quickly as Ralph powered home a header from Woods' corner.
Ariyibi and Turnbull both had shots saved as Motherwell got on top but Gorrin received a second yellow card following a reckless tackle on Robson.
Dundee nearly took advantage and Kusunga headed off the inside of the post from a Woods corner but Motherwell continued searching for a winner and Turnbull came close from 30 yards before Dieng stopped his powerful free-kick.
Jim McIntyre's men were handed a reprieve when referee Kevin Clancy withdrew a penalty award after initially penalising Kusunga for handball after Turnbull appeared to have kept the ball going over the byline. A goal-kick was given following the intervention of assistant referee Jordan Stokoe.
Motherwell looked the likelier team to score in the later stages but Dundee had chances to exploit the hosts' numerical disadvantage.
And Turnbull had the final say when he scored his 12th goal of the Premiership season after being set up by 17-year-old Semple minutes into the substitute's debut.