Having drawn 2-2 in Ukraine three weeks ago in the first leg, Borussia went through 5-2 on aggregate as goals by Felipe Santana, Mario Götze and Jakub 'Kuba' Blaszczykowski capped the German champions' dominant display.
After the defeat, Shakhtar coach Mircea Lucescu repeated his bold prediction that the winner of this tie will reach the final at Wembley on May 25 and said he hopes Dortmund prove him right.
"They have a strong team, they were deserved winners, and like I said yesterday, I think the team that won today will reach the final and I wish them plenty of success," said the Romanian.
Lucescu is not alone in backing Borussia.
Before his side crashed out at the hands of Real Madrid at home on Tuesday, Manchester United manager Alex Ferguson tipped Dortmund to win the title, but Klopp played down rising expectations.
"We have received many compliments on the way we have played in Europe and it feels good to win, but we can't look further than April 2 when the quarter-finals start," said the 45-year-old.
"There are still lots of strong teams in the competition. We are just delighted with the result and I am proud of the team."
Dortmund are in the quarter-finals for the first time in 15 years and can be joined in the last eight by Bundesliga rivals Schalke 04, who host Galatasaray, and Bayern Munich, who are home to Arsenal, next week.
Klopp said he had no preferred opponent in the last eight, whether or not that involved a trip outside Germany, or even just the 40 kilometre (24 miles) drive east to Gelsenkirchen-based Schalke.
"We'll have to travel where ever we go, well maybe not to Schalke, but we are just happy to be in the last eight," he said. "We'll watch the other games, see what the other teams do, it's just extremely cool to be in the quarter-finals."
This was their fourth straight home win in Europe this season and their fifth in all.
But the victory came at a cost with both defensive midfielders Sven Bender and Sebastian Kehl facing a race against time to be fit for Saturday's Ruhr derby at Schalke.
"Sven has an ankle injury, he's not sure to face Schalke, while Sebastian Kehl also aggravated a rib injury and we have to see how he goes," said Klopp.
The hosts dominated their Ukrainian opponents with Germany's rising star Götze, 20, and defensive midfielder Ilkay Gundogan proving a constant thorn in Shakhtar's side.
"We knew we had a huge chance to reach the quarter-finals and we delivered a great performance," said Götze. "Obviously scoring two goals in the first half gave us a lot of confidence."
Centre-back Neven Subotic said Dortmund are proving themselves on the European stage as they target their first Champions League win since 1997, when they beat Juventus 3-1.
"We hope it is our season, we have showed against a big team tonight and in a tough group that we can come out on top," said Subotic after Dortmund saw off Ajax, Real Madrid and Manchester City in the group stages. "We want to prolong this Champions League experience as long as we can."