Conor 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 Liverpool boss Klopp.
An Everton fan then 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 incidents overshadowed a pulsating clash between bitter rivals that could easily have finished with goals aplenty but for a man of the match display from Pickford and a similar performance from his Liverpool counterpart Alisson Becker.
"Wow, what can you do? It was a great derby, super intense, with a lot of really spectacular saves," Klopp said.
"We hit the post three times and Pickford made so many saves. I don't know how many derbies I've been in, but it's quite a few and they're always difficult here."
Everton are now without a victory in their first six games this term, while Liverpool's momentum was checked after winning their previous two matches.
"We kept our heads. We played well and we're getting better. It felt like today could have gone either way," Everton manager Frank Lampard said.
"We're not going to go from just staying up to being the team we want to be, we have to go through a process."
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Maupay finally made his Everton debut after signing from Brighton, but the French striker wasted an early chance when he scuffed wide from Demarai Gray's pass.
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.
The Reds' defence has been unusually creaky this season and, when Maupay caused panic inside the penalty area, Tom Davies seized possession and flicked a curling shot onto the far post.
After a subdued start from Darwin Nunez, Liverpool's summer signing from Benfica sparked into life with an agile effort that forced a brilliant tip onto the crossbar by Pickford.
With the Toffees 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 when Roberto Firmino was intorduced at half time, there was an immediate improvement in the away team's approach.
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.
Furhter saves followed from the England international, as he repelled Firmino's header, then made another superb stop to keep out Fabinho.
Coady's disallowed goal was the game's biggest flashpoint, leaving the home fans frustrated, and the drama continued when Alisson tipped Dwight McNeil's deflected shot over the bar.
Pickford was in an 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.
