Pritchard twice netted superbly in the first half to put Spurs in command but a penalty from Philip Hauck and a last-gasp Philipp Muller strike completed a dramatic late comeback from the Germans.
The result leaves the north Londoners tied with Wolfsburg on four points apiece in Group 1, with tough trips to Anderlecht and Barcelona still to come.
Tottenham took the lead just six minutes in when, having won a free kick with a superb solo run, Pritchard curled the set-piece sublimely past Niklas Klinger and inside the near post.
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The German side came even closer on 18 minutes when Avdijaj’s fierce shot deflected off Ball and crashed down off the crossbar before bouncing out.
On the half-hour mark Coulthirst spurned another presentable opportunity when a mix-up in the Wolfsburg defence gifted him possession with only Klinger to beat, but the Spurs striker tried to round the goalkeeper and saw his attempt smothered.
A minute later Coulthirst extended Klinger again, forcing the Wolfsburg stopper into a fine save with his legs with a low effort after good work from Kenneth McEvoy.
But Spurs were not to be denied for long and, on 33 minutes, Pritchard raced inside onto Milos Veljkovic’s incisive through pass before ripping a shot high into Klinger’s net.
Wolfsburg roared forward and, after a driving run and incisive pass from Kerem Bulbul, Jungbin Park forced Spurs goalkeeper Luke McGee into a good low save, allowing Spurs to carry their two-goal lead into the break.
In a scrappy but open start to the second half, Coulthirst went close again when he scuffed just wide from Laste Dombaxe’s chest down on the edge of the box.
Wolfsburg pushed forward, and on the hour mark Bulbul’s short free kick was met by Nils Winter, whose firm low strike was deflected just wide by the flying body of Kevin Stewart.
But the hosts continued to create the better chances, and Dombaxe should have put the game beyond reach moments later, skewing wide from inside the six-yard box after some skilful and tenacious work from Coulthirst.
Spurs kept coming and, soon after, Pritchard missed a golden chance to secure his hat-trick when he blasted McEvoy’s firm low cross over on the slide.
Wolfsburg made their opponents pay for their profligacy with 12 minutes left when Hauck was brought down by Jack Barthram inside the area. The midfielder picked himself up, dusted himself down and sent McGee the wrong way to set up a tense finale.
Spurs were no longer playing with the same assurance in the face of renewed pressure from the Germans, and McGee was forced into a sensational save in injury time to deny Federico Palacios.
But there was one more twist left in the tale as, fellow replacement Muller rounded the Spurs keeper a minute later and slotted in to complete a dramatic comeback.