The game appeared to be heading towards a frustrating draw until Matip’s late intervention.
After Klopp described last week’s 4-1 defeat in Napoli as a ‘horror show’, he demanded a response from his player.
Yet after Mohamed Salah’s first goal in eight European games had put them ahead, more defensive frailties allowed Mohammed Kudus, the striker who was seemingly on his way to Everton before team-mate Antony was sold to Manchester United, to equalise.
A laboured second half, which included a personnel and formation change, failed to produce any genuine threat until Matip rose highest at a corner and his header crossed the line despite the best efforts of Dusan Tadic.
It was the stroke of good fortune the Reds were in desperate need of as, although second-half substitute Darwin Nunez made an impact, the game appeared to be heading towards a frustrating draw until Matip’s intervention.
"I think everybody could see that we understood and that we had to put a completely different shift into the game. The boys did that,” declared Klopp.
"We played a lot of good stuff against a hard fighting opposition. Scored the first goal and should’ve scored more goals, from set pieces especially.
"I'm not sure it was their first attack, but Ajax didn’t have a lot. That’s how it is in our situation, anything can happen, but it was an incredible strike from (Mohammed) Kudus.
"It’s like another test. Everybody is hoping you go in the right direction and then you get another setback. You could see the impact of that. The pressing before the goal was much better than after the goal. Only a yard or two but if we press, we have to do it right.
"It is a first step, a very important step. Now we have a strange break, pretty long. Brighton is waiting."
Liverpool defender Virgil van Dijk has been questioned in recent weeks and he admitted the negativity that has been hovering over Liverpool has not affected his team-mates.
"We are not listening to the outside world,” he told BT Sport.
"A lot of ex-football players, who know what we go through, they say a lot to get us down, but we know the last game was very bad and this was a step in the right direction."
"It was very important to show a positive reaction. It's not easy to turn it around, but it is a step in the right direction.
"During the game there were great moments. It was very important to win going into the international break.
"I should have scored. We felt dangerous in the set-piece, especially corners, and it was our responsibility to convert one - thankfully Joel did.”