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Teale, who spent three months on loan at Plymouth from Derby last season, scored the Rams' first and laid on their third as Nigel Clough's men leapfrogged the injury-hit Pilgrims, who were without three first-choice defenders.
The Scotland international midfielder did not score for Argyle in 12 games and had netted only once before for Derby in 56 appearances, but gave the Rams the lead two minutes before half-time.
His first shot from 30 yards was blocked by Simon Walton, but the ball rebounded straight to him and he lashed it back over a flat-footed Romain Larrieu.
It was cruel luck on the Pilgrims, who had hit the woodwork twice before Teale's opener, once in the third minute when debutant Alan Judge's shot from a tight angle struck a post and rolled across the face of the goal.
Jamie Mackie then blasted a shot from 35 yards after a Robbie Savage error, but the ball hit the underside of the bar and hit goalkeeper Stephen Bywater before rebounding to safety.
However, the Pilgrims were architects of their own downfall after the break when mistakes by French defender Mathias Doumbe gifted Derby two goals.
The first came 25 seconds after the restart when Doumbe, normally a centre-back but filling in on the right side of defence, failed to clear and was dispossessed by Kris Commons, who teed up Hulse to squeeze the ball in from a difficult angle.
Six minutes later, Doumbe let in Teale, who made the by-line and cut into the penalty area before setting up Hulse for a tap-in.
The errors infuriated Argyle skipper Karl Duguid, who confronted his team-mate and had to be pulled away by Walton as the pair exchanged angry words.