The Toffees will leave Anfield feeling hard done by after Ashley Young was sent off before the break for two yellow card offences.
Dyche's ten men stood firm for much of they game and felt Liverpool should have been reduced to ten men in the second-half.
Konate, who was already booked, fouled Beto as he began to break for Everton but referee Craig Pawson didn't show a card.
It sparked angry protests from Dyche and he was shown a yellow, as was a member of his backroom staff.
Speaking on talkSPORT, former Liverpool midfielder Ray Houghton believes Konate got away with it: "He's a lucky boy."
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"Sean Dyche is fuming and I'm not surprised," added Kevin Campbell as Everton fans in the away end also expressed their frustrations.
Liverpool kept knocking at the door although their rivals held on with a one-man disadvantage, but only until the 74th minute.
Luis Diaz's cross hit the arm of Michael Keane inside the area and after a VAR review, Pawson awarded the penalty.
Salah, who had been unusually poor up until the closing moments, made no mistake from the spot and lashed in past Jordan Pickford.
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The Egyptian netted his second of the match in stoppage time to make it game, set and match for Jurgen Klopp and co.
It was clear that Young's dismissal played a role in Everton failing to get something out of the game.
They certainly have a claim that Liverpool should have also been sent down to ten men following Konate's challenge.
But it wasn't to be for the Toffees as they fell to their sixth defeat of the season, with their neighbours going top of the table.