But seven minutes into the half, cornerback Rudy Wormer was hit by Ivory Coast defender Simon Deli, who arrived in the club like Tau out of season.
It was a simple setup goal that gave Bruges a major rescue service after surviving severe pressure from a 32-degree heat in the Ukrainian capital.
Five minutes after half-time Brugge again fell behind in almost the same easy way, with Volodymir Shepelev finishing the effort.
Brugge then got a big break after Dinamo defender Mykyta Burd was sent off, reducing the hosts to 10 and taking the opportunity to go back over Wormer to 2-2.
Tau was then relegated to even numbers and in stoppage time, Kiev's hopes were suddenly revived after Brandon Mechele's own goal.
But with almost the last touch of the game, Louis Openda singled for Brugge to make it 3-3 and pass them in the play-offs.