Fatos Beqiraj rounded Oleksandr Shovkovskiy to create a crossing chance for the visitors on 25, but no-one could get on the end of it and the score remained 1-0.
Then on 32 minutes Kiev had their second. It was a series of defensive mistakes as first Kelava threw straight to Taye Taiwo before his cross was turned into his own net by Josip Pivaric.
It was the 49th minute when Zagreb managed their first shot on target, however Shovkovskiy was well behind the shot from range.
Miguel Veloso and Andriy Yarmolenko both squandered decent second-half chances for the hosts but the damage had been done in the first half and Kiev ran out 2-0 winners.