Diabate struck home a low shot in the 41st minute to give the Ligue 1 side a 2-1 aggregate victory.
However, the Ukrainians will no doubt rue their inability to translate their dominance, as well as a missed penalty, in the first leg into goals.
Niko Kranjcar had the first real shot on goal, curling a right-footed effort in from the left which was comfortably dealt with Bordeaux goalkeeper Cedric Carasso.
Gregory Sertic shot wide for the hosts but it was the Ukrainians who were showing more attacking verve, Miguel Veloso and Roman Bezus both missing with efforts.
Carrasso then prevented the visitors from going ahead midway through the first half when he parried Andriy Yarmolenko's cross-volley.
Veloso and Taye Taiwo both sent in shots from distance for Kiev but with the clock ticking down to half-time it was Bordeaux who took the lead.
Sertic found Diabate and he sent a low shot from close-range into the bottom right-hand corner.
Kranjcar, Oleg Gusev and Yarmolenko all had shots blocked as Kiev made their intentions clear at the start of the second half.
Lukman Haruna put a shot wide from close range and Yarmolenko had an effort blocked but Kiev were not creating anything of note as time wound down.
Domagoj Vida's injury-time header was comfortably held by Carasso as Bordeaux went through.