Burnley stood strong, but Liverpool's defensive frailties were highlighted again as Chris Wood bullied the two centre-backs and poked the ball in the direction of Arfield, who finished confidently into the bottom corner on the half volley on 27 minutes.
The former Dortmund manager was quick to heap praise on the performance of Philippe Coutinho as he returned to the starting line-up following intense speculation he would join Barcelona during the summer transfer window.
"I asked him to stand up, but then obviously when we got the drinks in this little interruption we saw".
"It was a disciplined performance but just that one goal".
The hosts quickly managed to regather shortly after via Mohamed Salah, but the scoreline remained unchanged with the result seeing Burnley jump past Liverpool in the Premier League standings.
Having appeared to announce itself as a potential title contender with a 4-0 victory over Arsenal in late August, Liverpool has slipped to eighth on the table, and Klopp will be concerned as to how his side will cope without Mane at Leicester in a week.
Liverpool welcome Burnley to Anfield in the Premier League on Saturday afternoon, and are big favourites to pick up all three points given the gulf in class between the two sides.
Liverpool were relatively untroubled for most of the game, but they always find a way to stumble in matches. The Liverpool front man does brilliantly to control Can's long ball, he brushes off a challenge before powering the ball into the back of the net from inside the box. "We were dominant, fantastic attitude".
But after winning at Chelsea and drawing at Tottenham, Sean Dyche's side can now add Anfield to the list of grounds where they have taken some reward in a superb start to the season.
"You know they're going to have some efforts but if you keep them as far out as we kept them, as in distance from the goal, you know the stats say it's much harder to score from 30 yards than it is from six". Let's hope they can make a few of these chances count in the second half.
"For me, it's a penalty, I can not change this even when I saw the pictures back after the game".
Liverpool failed to take advantage of their spells of dominance, particularly near the end of the game when substitute Dominic Solanke hit the crossbar.
"A lot of parts of the game I was really happy with".
"We have quality and now we have to use this quality consistently, that should be the plan", said Klopp.