The Premier League side had to dig surprisingly deep in sweltering Cologne, where Bruno Fernandes' spot-kick ended the Danish side's record European run in extra time.
The hard-fought 1-0 win seals a place in Sunday's semi-final against either Sevilla or Wolves, with Solskjaer determined to reach his first final as United boss having this season fallen in the final four of the Carabao Cup and FA Cup.
"But we've got six days. There's not many times that we have six days to prepare for a game."
While relief was the overriding feeling for those connected to United, counterpart Solbakken expressed enormous pride after Copenhagen pushed until the last.
"Of course I get annoyed when we were so close, but I cannot ask for more from the players," he said. "Every man has given himself completely, both physically and mentally, and we managed to live with the struggle right down to the very last."