The Hungarian midfielder gave Liverpool the lead after eight minutes of Sunday's home match when he silenced the travelling fans with a stunning 30-metre strike.
Jeremie Frimpong made it 2-0 when he cut in from the right and fired home a fierce left-foot shot.
But Szoboszlai offered Barnsley a way back into the game five minutes before halftime with a calamitous mistake that will feature in blooper reels for years to come.
His attempted backheel six metres in front of his own goal was poorly planned and shockingly executed and the chasing Adam Phillips only had to tap the ball into the empty net to make it 2-1.
"Sorry to the team," Szoboszlai told TNT Sports. "I made it hard for us with an easy mistake. Football happens and we move on."
The midfielder was also lucky not to give away a penalty early in the second half as Barnsley made a mockery of their League 1 status.
The Yorkshire club took the game to their more storied rivals and with the match still in the balance Liverpool coach Arne Slot was forced to call on his big guns, bringing on Florian Wirtz, Hugo Ekitiké and Ibrahima Konate after an hour.
However, much of the action continued to revolve around Szoboszlai.
He cleared a Phillips shot off the line at one end and then set up Wirtz moments later only to see him blaze the ball over the bar.
Wirtz made amends six minutes from time when, with the visitor's energy levels dropping, he took an Ekitiké pass and curled a lovely shot into the far corner.
Ekitiké made it 4-1 in stoppage time when he tapped home a Wirtz cross.
It was an unfortunate end for a visiting team who acquitted themselves admirably against a team 57 places above them in the league pyramid.
Although the scoreline flattered the home side the result was Liverpool's 11th straight game without defeat and a sign of its ability to grind out results.
Liverpool have not lost since November to Paris Saint-Germain in the Champions League.
Brighton will come to Anfield on the weekend of February 14.
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