Liverpool s Mohamed Salah reacts at the end of the English Premier League soccer match between Everton and Liverpool at Goodison Park, Liverpool, England, Saturday, Sept. 3, 2022. (AP)
Coady thought he had put Everton ahead with a close-range finish from Neal Maupay's shot in the second half at Goodison Park.
But the former Liverpool defender's goal was correctly disallowed for offside after a VAR check, sparking an angry response from the stands.
A beer bottle appeared to be thrown from behind the managers' technical areas, narrowly missing Klopp.
As if that wasn't shocking enough, an Everton fan ran on to the pitch from the Gwladys Street End in a bid to confront referee Anthony Taylor before being ushered away by stewards.
Those nasty incidents overshadowed a pulsating clash between these bitter rivals that could easily have finished with goals aplenty but for a man of the match display from Everton keeper Jordan Pickford and some fine stops from his Liverpool counterpart Alisson Becker.
Everton's 2021 triumph at Anfield remains their only win in the last 28 Merseyside derbies in all competitions.
Frank Lampard's team are now without a victory in their first six games this term, while Liverpool's momentum was checked after winning their previous two matches.
Maupay was finally making his red-tape delayed Everton debut after signing from Brighton, but the French striker wasted an early chance when he scuffed wide from Demarai Gray's pass.
Tom Davies had started that attack with a crunching challenge on Fabinho that epitomised the derby maelstrom and left the Liverpool midfielder pole-axed for several moments before he could play on.
As the intensity increased, Luis Diaz had Everton fans baying for blood as the Liverpool winger went unpunished after catching Nathan Patterson with an out-stretched arm.
Liverpool striker Darwin Nunez looked rusty in his return from a three-game suspension, failing to guide his far-post header on target from Trent Alexander-Arnold's cross.
Keeping Anthony Gordon despite Chelsea's sustained interest was Everton's most significant achievement so far this season and the young forward gave a glimpse of his talent with a stinging shot that brought Alisson into action.
- Frenzied tempo -
Liverpool's defence has been unusually creaky this season and, when Maupay caused panic inside the penalty area, Davies seized possession and flicked a curling shot onto the far post.
After a subdued start from Nunez, Liverpool's big-money signing from Benfica sparked into life with an agile effort that forced a brilliant tip onto the crossbar by Pickford.
With Everton struggling to regroup, Diaz pounced on the rebound and stroked a curling strike against the post.
Liverpool had found it hard to impose themselves amid a fiery derby played at a typically frenzied tempo.
But Klopp sent on Roberto Firmino at half-time and there was an immediate improvement in Liverpool's approach play.
Kostas Tsimikas fired high and wide from a good position in the six-yard box, but Liverpool were hitting their stride.
Nunez's dipping volley tested Pickford from the edge of the area before Firmino's low drive was pushed away for a corner by the Everton keeper.
Pickford was all that stood between Liverpool and the lead as he repelled Firmino's header, then made another superb stop to keep out Fabinho's blast.
Liverpool almost paid for those misses when Gray teed up Maupay six yards from goal, but the striker's poor touch allowed Alisson to get in position to save.
Coady's disallowed goal was the game's biggest flashpoint, leaving Everton fans howling in frustration.
There was still more drama as Alisson made a stunning save as he tipped over Dwight McNeil's deflected shot.
Pickford was in equally defiant mood, turning away Firmino's snap-shot in the final minutes and then pushing Mohamed Salah's shot onto the woodwork in stoppage-time.