Proper struck on the run from the edge of the area in the 86th minute, six minutes after Luuk de Jong had revived home hopes with his 19th goal of the season. They had gone behind to Sergei Ignashevich's penalty in the 76th.
The win left PSV in second place on 10 points behind group winners VfL Wolfsburg, who have 12 after winning 3-2 at home to Manchester United. The English club finished on eight points and have been relegated to the second-tier Europa League.
PSV, who knew they would qualify with a win, or a draw if United lost to Wolfsburg, are the first Dutch club to reach the Champions League knockout stage in the last eight years. CSKA finished bottom with four points from their six games.
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For a brief spell the Dutch champions looked to be headed for last place in the group after conceding a penalty that allowed the Russian army club to go in front.
But CSKA's lead lasted two minutes before De Jong showed his predatory instincts, taking the ball down in the penalty box and turning towards goal before calmly slotting home.
Propper's winner meant PSV were masters of their own destiny although they had some anxious moments before the final whistle.
It had taken almost half-an-hour to produce the first real effort of the night despite the home side probing frequently, especially down the left hand side of their attack.
De Jong had a close-range header saved in that first dangerous attack when he rose above two defenders to get on to the end of a cross from Hector Moreno.
A curling free kick from Jeffrey Bruma sailed just over the bar 10 minutes into the second half before CSKA winger Ahmed Musa brought a sudden silence upon the Philips Stadium as he whistled a stinging effort narrowly wide in the 61st minute.
A deflection from the PSV defence provided Musa with a snap effort that was perilously close to the opening goal.
The CSKA penalty came after a clumsy attempted clearance from Andres Guardado, who caught the leg of Zoarn Tosic right in front of Spanish referee David Fernandez Borbalan.
It threatened to turn PSV's night on its head but it was not long before the stadium was celebrating a decisive turnaround.
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