Liverpool's first goal of the game came under controversial circumstances when the Reds were awarded a penalty after Salah's shot hit Robin Koch's arm.
Any outstretched arm is enough for a referee to award a penalty but replays showed the ball had hit Koch's leg first which cast some doubt over why the decision wasn't reviewed by VAR.
Salah stepped up to take the penalty and buried it down straight down the middle of the goal.
Leeds, on their return to the Premier League after 16 years away, didn't have to wait long for their equaliser though with Harrison finding the back of the net three minutes later.
Liverpool fell foul to offside on 16 minutes when Sadio Mane was adjudged to be ahead of the Leeds defence but the Premier League champions rectified the scoreline four minutes later when Leeds conceded a corner.
Van Dijk was left unmarked and the force of his header forced Illan Meslier tumbling back into his own goal for Liverpool's second.
The Liverpool centre-back was at fault, however, on 30 minutes when Bamford pulled back another goal for Leeds. Stuart Dallas put a speculative ball towards the Liverpool goal and Van Dijk, not realising Bamford was behind him, flicked it straight into the striker's path. He struck it past Alisson to bring the score to 2-2.
Salah grabbed Liverpool's third and the fifth goal of the game on 33 minutes after Robertson's free kick was cleared only as far as the Egypt international. Completely unmarked, he took his time controlling the ball before belting it into the roof of the net.
Liverpool came out after half-time the brighter side with a strong shot from Georginio Wijnaldum well saved by Meslier on 49 minutes.
Liverpool failed to capitalise on several good chances and instead it was Leeds who pulled a third goal back on 66 minutes.
Klich and Costa combined to work the ball up the pitch before the midfielder perfectly volleyed the ball past Alisson.
It looked like Van Dijk has scored his second on 78 minutes after striking the ball into the net from a corner but the goal didn't count as Curtis Jones had fouled his marker in the lead up.
"It was so tough. It was a good start for us," Salah said after the game. "Tough team. The game for intensity was really high.
"I scored a penalty in a Champions League final so that was a little bit easier. It was just important for me to stay focused."
Leeds head coach Marcelo Bielsa told the BBC that he was happy with how his side matched the champions.
"Liverpool made it hard for us to neutralise them," he added.
"In the second half especially we struggled to get the ball from defence to attack.
"We also stopped Liverpool playing like they usually do and we made a lot of effort. During large parts of the game we were able to match them."
Liverpool manager Jurgen Klopp said he "loved" the spectacle of the game and felt like his side left space for improvement in their first game back.
"The opponent forced us to make mistakes, we can do better, we will do better but I loved a lot about the game against a well-organised, passionate side like Leeds," he added. "We used our skills to cause them problems, we could have scored more and in the end we used set-pieces which is fine by me.