Wanderers had not won in front of their own fans since September and finally broke the barren spell - albeit through a sliced own goal from visiting striker Marlon King.
Solbakken said: "It's a lot of relief (to end our home run without a win).
"A lot of things were maybe put to bed today - a clean sheet, a home win, a good second-half (performance) and a derby win. We have two very difficult away games coming up so it was important game in many aspects.
"I think we should concentrate on our performances and not look so much at the table. It's still not good reading."
Wolves leap-frogged Birmingham with the win and now sit in 16th place in the Championship.