The win gave Greece, who have reached the finals for a second time after making their bow in 1994, a 1-0 aggregate victory after their European playoff first leg ended goalless in Athens four days ago.
Salpigidis struck on 32 minutes when he beat keeper Andriy Pyatov from close range.
Ukraine had their opponents on the back foot at the start, boxing them into their own half as the Greeks initially laboured.
The hosts came close through Andriy Shevchenko in the eighth minute when the striker first failed to connect with a low cross in the box and then fired wide from six metres.
Greece responded with a chance of their own when Giorgos Samaras rose above the Ukrainian defence to head a Giorgos Karagounis free-kick over the bar.
The 2004 European champions then struck against the run of play when Samaras sliced open the Ukraine defence with a pass for Salpigidis to fire home.
The Greeks were lucky not to concede an equaliser just before the break when Ukraine defender Evgen Khacheridi, who had ventured forward, deflected a cross intended for Shevchenko wide.
Greece, buoyed by their lead, became more confident in the second half, tackling harder and finding more space in attack as the hosts surged forward in numbers.
With the clock ticking and in need of two goals, Ukraine came close to equalising with substitute Evgen Seleznev firing just wide on the hour.
Another effort by Yaroslav Rakytsky minutes later was blocked by Greece keeper Alexandros Tzorvas.
The final whistle found Greeks cheering in front of almost empty stands as most of the disappointed 30,000 Ukrainian fans had left before the end.
