Wilkinson said in Monday's news conference that she had been generally disappointed with first-half performances since she took charge, but she would have been pleased with how the game started.
Wales were almost ahead in the first minute when Gemma Evans' long ball played through Rachel Rowe, but Darya Keliushyk was equal to the effort.
The chances came thick and fast, with three corners in the opening stages all causing the Ukrainian defence problems.
The best chance from those set pieces fell to Kayleigh Barton, who was left completely unmarked at the front post, but could not get her header on target from 10 yards out.
Barton almost made amends after 15 minutes, when she got in behind the Ukraine defence and chipped the ball over Keliushyk, but Lyubov Shmatko got back to stop the ball on the goal-line and clear.
Despite dominating the game, it was Ukraine who took the lead just as they had done last week.
The Welsh defence failed to pick up the run of Olha Ovdiychuk, and her low ball across the box was dummied superbly by Roksolana Kravchuk which left Yana Kalinina with a tap-in.
The second half started much like the first half, with Wales fully in control of the game and Ukraine boxed into their own half.
Ceri Holland almost scored an accidental equaliser two minutes after the restart, as Keliushyk was forced to tip her goal-bound cross on to the top of the crossbar and behind for a corner.
If the match appeared to be following a similar pattern to Friday’s game, things got even more alike when Wales were awarded a penalty as Barton bundled to the ground.
And as she did at Parc y Scarlets, Barton scored the resulting spot-kick, thumping the ball down the middle.