Milevskiy struck the woodwork again moments before the half-time interval and Yaroslav Rakitsky forced James to make a spectacular save after the break as Ukraine continued to dominate.
Andriy Yarmolenko and Artem Milevskiy both failed to test English goalkeeper Joe Hart when well-placed, a failing that would become a feature of Ukraine's play in the match.
Co-hosts Ukraine saw a goal not given during a must-win group-stage match with England, when John Terry cleared Artim Milevskiy's shot after it looked to have crossed the line.