The Ukrainians were the top seeds in the draw but were in fourth place going into Thursday's game after taking just one point from matches against Genk and Rapid Vienna.
But Andriy Yarmolenko's stunning first-half strike set them on their way and Dieumerci Mbokani and Oleg Gusev made the game safe in the second period.
Jeremain Lens shot wide before Yarmolenko sounded the first serious warning to Thun with a left-foot shot against the crossbar from 25 yards.
Yevgen Makarenko was off target from distance and Younes Belhanda sent a shot over before firing a free-kick into the wall.
The inevitable breakthrough arrived, though, when Danilo Silva teed up Yarmolenko 25 yards out and he tricked his way past Sekou Sanogo before firing powerfully past keeper Guillaume Faivre with his left foot.
Thun finally responded but Dennis Hediger's long-range attempt did not threaten the goal and they remained behind at the break.
Thomas Reinmann blocked Mbokani's effort and Yarmolenko drew a point-blank save from Faivre as Thun tried to keep themselves in the game.
But Mbokani was not to be denied and headed in Danilo Silva's cross at the near post in the 59th minute to double Dynamo's advantage.
They quickly went in search of a third but Lukman Haruna shot wide and Lens missed two chances.
Thun's Christian Scheuwly could not make the most of a free-kick chance and sent another effort wide before the hosts scored their third in the 78th minute.
Belhanda broke following a Thun free-kick and Serhiy Sydorchuk squared the ball from the left of the penalty area for Gusev to slot home.
Mbokani saw a free-kick saved and then headed wide before Faivre brilliantly saved Belhanda's free-kick to keep the scoreline relatively respectable.