Jack Hamilton was called into action early and pulled off a double save to keep the Ukraine attack out in the sixth minute.
Ukraine opened the scoring in the 26th minute when Dmytro Khlyobas hit the back of the net.
After the break, Andrew Robertson linked up with Henderson but the Hibernian midfielder failed to hit the target.
Igor Duts scared Hamilton with 20 minutes remaining when his long-range effort shaved the crossbar, but Ukraine made no mistake with their next attack.
The ball fell to Oleksandr Svatok inside the penalty area and he unleashed a fierce strike into the back of net.
The woodwork denied Cummings the chance to win the match in the final five minutes when his shot rebounded off the crossbar.
Dom Hyam saw red in stoppage time after pulling down the Ukraine striker when one on one but the visitors were unable to take advantage.