Aberdeen manager Stephen Robinson had spoken pre-match about the need to keep going until late in the game, which had paid dividends in the comeback win over Hearts the previous week.
Robinson would have been hoping not to face another deficit but the signs were there even before the Westley goal, with 22-year-old Bevan afforded too much space to get a shot away that flew wide early on.
It was a warning the visitors did not heed, with Bevan laying on the opening goal and consistently getting into dangerous positions on the right.
Through that outlet, Dundee should really have added to their lead in the first half. Bevan flashed the ball across the face of goal from an advanced position, with Westley waiting in the middle, then tested Muller with a firm strike.
But, after something of a rally from Robinson's visitors earlier in the second half, the winger effectively finished the contest, showing poise and power with his clinching goal.
Hay's strike after chasing a simple long ball, and a header from Finlay Robertson's cross, gave Muller more to do on a day when he was by far Aberdeen's best player.
Though more forceful the longer the game went on, the damage had been done for Aberdeen, with their promising start to the season coming to an abrupt halt.