Wolves needed a penalty shoot-out to progress - Jordan Roberts seeing the 14th spot kick saved by Wolves' goalkeeper Carl Ikeme at Molineux on Saturday.
Wolves had taken a 53rd-minute lead through Sylvan-Ebanks Blake only for Michael Rankine to equalise nine minutes later.
Former Cologne boss Solbakken said: "Some of our stuff was good and some not so good.
"We struggled to open them up but it was a deserved victory when you count the chances.
"The fans should expect that we are going to be a good team at home.
"But we lacked penetration in the last third of the pitch.
"That means that in some periods of the game I was disappointed we didn't create more chances.
"But if we had kept the 1-0 for another five minutes it could have been two, three or four. But the cup is about going through."
Wolves broke the deadlock when new signing Tongo Doumbia, who is on a season-long loan from Rennes, picked out Ebanks-Blake whose rising drive flew past goalkeeper Jamie Young.
But any hopes Wolves had that one goal would be enough to see off Aldershot disappeared nine minutes later when Rankine rose at the far post to head home a corner from Josh Payne.
Alex Rodman had two excellent chances to earn Aldershot a shock win in extra time before the game was decided by penalties.
Ebanks-Blake, Stephen Hunt, Stephen Ward, Roger Johnson, Karl Henry, Frank Nouble and Anthony Forde all scored for Wolves.
Josh Payne, Craig Reid, Rankine, Alex Rodman, Ben Herd and Aaron Morris were on target for Aldershot before Roberts was denied by Ikeme.
Aldershot manager Dean Holdsworth felt that the defeat was hard on his players who more than held their own against a team two divisions above them.
He said: "I feel disappointed for the players who gave a very good account of themselves.
"I think we have won when we have lost really. I am quite proud of that performance.
"We know the qualities of Wolves but our game plan was working, we got ourselves back in the game and I am delighted with a lot of aspects of that.
"We won't play Wolves every week in Division Two so there are a lot of positives from that performance. That is a platform for us now.
"We have got to be realistic in what we can achieve this season and a play-off spot would be fantastic.
"I want to go about things very quietly and I think we will do all right.
"We have a good bunch of players and we know our strengths and we will have a real good go this year at trying to get in the play-offs.
"They are a very young squad but today will give them a lot of experience and enjoyment."