Dortmund dominated the encounter at Signal Iduna Park from start to finish, with goals from Felipe Santana, Mario Gotze and Jakub Blaszczykowski sending the Bundesliga champions through to the last eight by a 5-2 aggregate scoreline.
And the quality of the performance was not lost on Klopp, who told Sky Sport: “This is an extraordinary moment, and it feels like an extraordinary moment.
“We saw during the second half how strong Shakhtar can be if you let them play. But we played great for 80 minutes and did not let their offence come into the match.”
The coach’s sentiments were also echoed by Gotze, who was impressive throughout the match, scoring one and setting up another during the first half.
“It was an amazing match, and we had great chances,” the 20-year-old told reporters.
“The first two goals made us very calm, and after the third goal, we played magnificently.”
Dortmund, who are in action next in the Bundesliga for a Revierderby against Schalke on Saturday, join Real Madrid as the first two teams in the draw for the Champions League quarter-finals.