Bales free-kick was headed into the Ukraine goal by visiting captain Andriy Yarmolenko in the 34th minute in the decisive moment in Cardiff.
The victory ends Ukraine’s emotion-charged bid for World Cup qualification amid their war with Russia, but its an historic moment for Wales who hadn’t reached the finals since 1956.
“It’s the greatest result in history for Welsh football,” Bale told Sky Sports.
“It means everything. It’s what dreams are made of. I’m speechless because I’m so happy.
“Words can’t describe how we feel at the moment.”
Veteran Wales goalkeeper Wayne Hennessey was outstanding with a string of top-drawer saves.
The triumph marks another major milestone for this generation of Welsh players, who reached the Euro 2016 semi-finals.
“I’m so proud of the guys, they thoroughly deserved it, Wales boss Rob Page said.
“The one thing these boys were missing was a World Cup and now we’ve got there.”
Wales will be pitted in Group B with England, Iran and USA in Qatar.