Klopp 'can imagine frustration' for Dyche after red card controversy in derby win
Even Liverpool boss Jurgen Klopp said he could understand Everton's frustration at a lack of consistency from referee Craig Pawson.
Ashley Young was sent-off before half-time for two bookable offences, while Ibrahima Konate escaped a second booking in the second-half with the scores still level.
Liverpool toiled to make their man advantage count, but finally achieved the breakthrough 15 minutes from time when Salah smashed home from the penalty spot after Michael Keane handled.
Salah was then teed up by Darwin Nunez to make the points safe seven minutes into stoppage time.
Victory takes Jurgen Klopp's men above Tottenham and Arsenal at the top of the table on goal difference with both sides from north London in action later this weekend.
"I can imagine the frustration of Everton and Sean (Dyche) in this moment absolutely," said Klopp on Konate escaping a red card 25 minutes from time.
The Reds have lost just one of their last 29 Merseyside derbies and remain unbeaten at Anfield in front of a crowd against Everton since 1999.
No fixture has had more red cards in the Premier League era that clashes between these two and another sending off proved decisive.
Everton had frustrated the hosts and posed a threat themselves until the 38-year-old Young showed his age rather than his experience by diving in to chop down Luis Diaz and earn a second yellow card on 37 minutes.
Klopp had expressed his frustration at being selected for the Saturday lunchtime kick-off just hours after some of his stars returned after playing for their countries in South American World Cup qualifying.
Colombia's Diaz and Argentina's Alexis Mac Allister started, while Nunez was introduced off the bench after his starring role in Uruguay's 2-0 win over Brazil in midweek.
Liverpool looked jaded despite having the extra man in the second-half.
Everton boss Sean Dyche introduced Keane at half-time to revert to a back five and the visitors restricted Liverpool to few clear chances to the frustration of the home support.
"We had to get used to that it took a little bit too long for my taste, but the important thing is we didn't get frustrated," added Klopp of making the man advantage count in the second period.
Dyche stunned
The Toffees were then stunned that Konate was not also given his marching orders as he pulled down Beto after being shown a yellow card.
To rub salt in Everton wounds, Klopp immediately replaced the Frenchman with Joel Matip and Dyche saw a yellow card for taking his protests too far.
"I have no clue how he didn't feel it was a bookable offence," said Dyche. "I like to think there are a lot of fair-minded people here today who are stunned that wasn't a second yellow card."
Liverpool have had their own troubles with officials of late after Diaz had a goal in their only defeat of the season at Tottenham last month wrongly ruled out following VAR confusion.
This time VAR came to their rescue as Diaz's cross struck Keane's outstretched arm.
After a pitch-side review, the referee pointed to the spot and Salah scored his seventh goal of the season to muted celebrations.
As Everton pushed forward in search of an equaliser, they were caught on the counter-attack deep into stoppage time as Nunez timed his pass to Salah perfectly for a simple finish.