But Liverpool weathered the storm to claim the win and end a run of two Premier League defeats, leaving them eighth on 19 points after 13 games. Spurs slipped to fourth on 26 from 14.
"I think we were lucky in the first half, we played quite good and managed to score two goals," Salah said. "In the second half we could've scored but were unlucky.
"We did well to get the three points. It is always a tough opponent. I'm focused on making a difference and managed to do so today."
Four minutes later, Nunez was at it again with a right-foot effort that flew past Lloris's far post.
Liverpool then took the lead when Salah controlled a pass from Nunez inside the area, turned and expertly fired a low shot into the corner with his left foot.
Spurs were under pressure, but four minutes later they almost equalised when Kane broke down the right and crossed to the impressive Perisic whose header hit Alisson and then the post.
Salah was handed his second goal on a plate after 40 minutes when Dier's attempted header came off his shoulder from a long kick by Alisson and the Liverpool frontman dinked the ball past Lloris.
Tottenham started the second half much brighter, as they usually do, and Perisic was desperately unlucky to hit the bar when he met a cross to the far post from Dier.
A huge cheer greeted Sweden international Kulusevski when he made his first appearance for seven weeks in the 68th and two minutes later he set up Kane who fired home to rally the Spurs fans.
But despite a late Tottenham onslaught, including a close-range header from Rodrigo Bentancur, Antonio Conte's side could not find an equaliser and Liverpool left with the points.
"It has been the story of our season so far -- we haven't played well enough in the first half and got punished," said England striker Kane.
"We fought until the end and we knew if we got one back we would have chances to get a second but unfortunately we couldn't do it and it's disappointing."