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United, trailing 2-0 from last week’s first leg, gave themselves hope through Anthony Martial’s penalty but Coutinho struck with the last kick of the first half, brilliantly deceiving goalkeeper David de Gea at his near post, as Liverpool reached the quarter-finals.
Klopp told BT Sport Europe: “They tried to play really offensively, Man United played good, but with our chances it was a little bit Liverpool against De Gea.
“The end of the first half, it was not the biggest chance we had but the skill of Coutinho was amazing, the game changed but it was intense until the end. It was not too easy so I’m pretty proud of the boys.”
Dejan Lovren and Mamadou Sakho stood up well to United’s pressure and Klopp said: “They had a lot of close situations and had to find some solutions – legal solutions – it is not so easy.”
Looking ahead to the quarter-finals, Klopp has an eye on his former club Borussia Dortmund – 5-1 victors on aggregate against Tottenham – but said: “I have contacts with my Dortmund friends, they all would like to play against us – but in Basle (in the final).
“But you saw all the results, there are only good teams left in the race.
“It’s cool. Last eight? Perfect.”
Opposite number Louis van Gaal was frustrated to have conceded so close to the interval.
“It’s very disappointing because we created enough chances,” he said.
“We know when we are attacking in this way that Liverpool also have counter chances. The way we conceded that goal, in the last seconds of the first half… you could have built up the second half so beautifully but at half-time they were also disappointed.
“In the second half we also had chances, (Marouane) Fellaini with his head and his foot – but you have to score three goals and that’s not too easy. If we did not concede that goal it was possible, I think.”
Van Gaal refused to criticise the otherwise impressive De Gea for his efforts on the goal.
“No, you have to also give credit to Coutinho,” he said. “It was a fantastic action.
“You can also say we had fantastic actions, but you need also a little bit of luck – but when you say that too much as manager, it’s not good.”