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Follow UsJurgen Klopp was shown a yellow card for running towards the fourth official and celebrating Liverpool's 94th-minute winner in the 4-3 victory at Anfield.
"For the way he acted on the touchline, he should be banned from the technical area," Sutton said.
"I don't think a fine is enough. He's got previous."
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The Liverpool boss, who injured himself in running towards fourth official John Brooks, later said the way referee Paul Tierney spoke to him when he received his booking was "not OK".
But Referees' body the PGMOL said it "strongly refutes" Klopp's suggestions that Tierney's actions were "improper".
Speaking on BBC Radio 5 Live's Monday Night Club, Sutton added: "It's a big deal. There will have been coaches and kids watching and thinking it's OK to run towards and abuse officials.
"As far as I'm concerned, Jurgen Klopp should be banned. He knows what he did was wrong."
Martin Cassidy, chief executive of Ref Support UK, echoed Sutton's call for Klopp to receive a touchline ban.
The charity has also called on the Football Association to consider more severe punishments to deter such behaviour in future - including points deductions.
"We believe the ban should be at least three games," Cassidy told BBC Sport.
"We also believe the FA should look at bringing in points deductions at all levels for those who regularly abuse match officials."
Jurgen Klopp was given a one-match ban after he was sent off against Manchester City in October for berating the referee's assistant.
The German also received a £30,000 fine, in addition to being warned about his future conduct.
The Football Association has been told to hit Liverpool with a points deduction due to Jurgen Klopp's behaviour during his side's 4-3 victory over Tottenham Hotspur.
Klopp celebrated Diogo Jota's injury-time winner in the face of fourth official John Brooks before making disparaging post-match comments which included accusing referee Paul Tierney of lying and having a problem with Liverpool.