Leading through Sadio Mane‘s brilliant second-half strike, the Reds conceded an 87th-minute equaliser when substitute Sam Vokes flicked on a cross for Johann Berg Gudmundsson to nod in unmarked from three yards.
However, Ragnar Klavan – one of the playsers most under threat after Van Dijk’s arrival was officially confirmed on Monday – was the unlikely match-winner when he popped up at the far post to nod in from close range.
“I was never in doubt about the character of my boys,” said Klopp.
“I never thought we had a defence problem or thought we need an extra player though Virgil will help us a lot.
“We are fourth in the table, absolutely OK. We need to keep the distance not bigger to the teams ahead of us, if we can make it smaller then it is better.”
With three of Liverpool’s ‘Fab Four’ absent – top scorer Mohamed Salah (groin) and Philippe Coutinho (groin) injured and Roberto Firmino rested – more responsibility was placed on Mane and he delivered.
It was only his second league goal since August but despite injury worries over two of his star players Klopp is happy to allow the Senegal international – and Salah – to fly Ghana for the Confederation of African football Player of the Year awards on Thursday, 24 hours before a Merseyside derby FA Cup tie at home to Everton.
“We have everything organised. We have two of the three best African players in our squad,” added Klopp.
“We have to show them respect. If it had been a matchday they wouldn’t have asked. We sleep in a hotel, they sleep in a plane. That is all the difference.
“If Mo is not fit, it will not be for that. For our side, it is just a sign of respect. I had (Pierre-Emerick) Aubameyang (at Borussia Dortmund) and there is nothing more important than to go there.
“Sadio is unbelievably important for us. He is strong, quick, so many things. He scores a goal like this and it is like ‘Wow’.
“He comes into the dressing room and the whole team gives him a hand. We need him, we need him so much for all teams coming.
“I am happy because I had a lot of players with moments where it wasn’t clicking and he still scores.”
Manchester United striker Romelu Lukaku will be sidelined for just a single week following his head injury, according to Old Trafford boss Jose Mourinho on Monday.
Lukaku, who has scored 15 times this season, was stretchered off against Southampton on Saturday following a clash of heads with Wesley Hoedt.
That ruled Lukaku out of Monday’s trip to his former club Everton and he is set to be absent for the FA Cup third round tie with Derby on Friday.
Speaking prior to the Everton match, Mourinho told Sky Sports: “I would say a week (out for Lukaku).
“It’s nothing really important – just a little protection, but no problem.”
United’s next Premier League game, at home to Stoke, isn’t for 10 days after the Derby game, giving Lukaku time to return for that fixture.
Lukaku crashed into Southampton’s Wesley Hoedt on December 30 as the pair contested an aerial challenge early in the first half at Old Trafford.
The Belgium striker lay prone on the pitch barely moving while he was treated by medical staff for five minutes.
Lukaku remained conscious and was eventually carried off wearing an oxygen mask.
Centre-back Ragnar Klavan was Liverpool’s unlikely match-winner at Burnley as his close-range header in the fourth minute of added time made light of the absence of three of the ‘Fab Four’.
With £75million world-record defender Virgil van Dijk waiting in the wings for his debut, the focus has been turned on the Reds’ central defenders, but it was at the other end of the pitch where Estonian Klavan proved his worth.
Jurgen Klopp’s side looked like they had thrown away two points when Johan Berg Gudmundsson’s 88th minute equaliser cancelled out a stunning strike from Sadio Mane.
However, Klavan popped up barely a yard out to nod in from a free-kick move to secure a hard-fought 2-1 win.
The midfielder made his first Premier League start of the season and was the busiest player on display, making a potentially goal-saving tackle at one end and conducting proceedings at the other.
Highlight of the match
Mane’s brilliant one touch and left-footed strike into the roof of the net for the match-winning goal.
Victory was only Jurgen Klopp’s fifth in 19 competitive matches in January since he took charge at Liverpool.
Nick Pope 7
Phil Bardsley 6
Ben Mee 6
James Tarkowski 6
Charlie Taylor 6
Scott Arfield 7
Steven Defour 7
Jeff Hendrick 5
Jack Cork 5
Johan Berg Gudmundsson 6
Ashley Barnes 7
Sam Vokes (for Hendrick, 71) 7
Nahki Wells (for Arfield, 86) 6
Simon Mignolet 6
Trent Alexander-Arnold 7
Dejan Lovren 5
Ragnar Klavan 6
Joe Gomez 6
Alex Oxlade-Chamberlain 7
Georginio Wijnaldum 6
Emre Can 5
Adam Lallana 7
Sadio Mane 7
Dominic Solanke 5
Roberto Firmino (for Mane, 72) 6
James Milner (for Lallana, 86) 6
Joel Matip (for Oxlade-Chamberlain, 90) 5