The Premier League champions were toothless in the first half but finally clicked 10 minutes after the break when the impressive Trent Alexander-Arnold set up Diogo Jota for a tap-in.
Liverpool edge past Midtjylland but defensive woes deepen
Lacklustre Liverpool beat Midtjylland 2-0 on Tuesday to make it two wins out of two in the Champions League but their defensive problems mounted as Fabinho limped off injured.
A late penalty scored by substitute Mohamed Salah gave the scoreline unwarranted gloss.
The manager left his first-choice attack of Salah, Sadio Mane and Roberto Firmino on the bench against the Danes, giving his second string a chance to shine.
But Liverpool, lacking urgency and fluency at an empty Anfield, struggled to break down the hard-working visitors, who lost their Champions League opener to Atalanta 4-0.
The opening could have been worse for Klopp's men.
Midtjylland had a chance to take a shock lead in the opening minutes when a long clip down the middle sent Anders Dreyer in on goal and Alisson Becker had to make a smart save.
Liverpool, who beat Ajax away in their Champions League opener, hogged possession but struggled to create clear chances in the early stages.
Most of what little threat there was in the first half came from full-back Alexander-Arnold, who produced a cross which Takumi Minamino failed to connect with.
Alexander-Arnold then danced through the Midtjylland defence midway through the opening period but failed to find the Japan international.
But Klopp's problems became a lot more serious just before the half-hour, when Fabinho, filling in alongside Joe Gomez in central defence, limped off with an apparent hamstring injury.
Klopp is already coming to terms with the absence of central defensive lynchpin Virgil van Dijk, who could be out for the whole season.
As half-time approached, Alexander-Arnold drilled another cross to Minamo, who headed wide.
But Liverpool failed to register a single first-half shot on target for the first time in 51 home games in all competitions, since October 2018.
Gini Wijnaldum replaced captain Jordan Henderson at half-time and after a few minutes Klopp ordered Salah and Mane to warm up on the sidelines.
The breakthrough finally came when Alexander-Arnold received the ball on the right and exchanged passes with Xherdan Shaqiri before crossing for Jota to finish from close range.
It was Liverpool’s 10,000th goal in all competitions, 128 years after their first, netted by Jock Smith in September 1892.
Salah and Mane came on just after the hour mark for Minamino and the disappointing Divock Origi but Liverpool still struggled for rhythm.
Firmino missed a clear chance for Liverpool in the closing minutes after yet more good work from Alexander-Arnold.
They almost paid the price immediately but Dreyer could only find the side-netting after drifting past Gomez.
Liverpool escaped and Salah won the penalty which he converted.
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