The Egypt international has signed a five-year contract for a fee in the region of €39m, including add-ons.
The current exchange rate means that would work out at £34.3m — not quite enough to break the club-record £35m paid to Newcastle for Andy Carroll in January 2011.
Salah was a long-term target of the club, who previously tried to sign him three years ago when they were pipped by Chelsea, and current manager Jurgen Klopp was keen to add more pace to his side.
The club had a £28m offer turned down earlier this month but after finalising terms with the Serie A club on Wednesday, Salah travelled to Merseyside to complete his medical and sign his contract.
“I’m very excited to be here. I’m very happy,” he told liverpoolfc.com. “I will give 100% and give everything for the club. I really want to win something for this club.
“Everyone can see the coach gives everything. I hope to see that together we can give everything to win something for the club, for the supporters and for us.”
Salah, who scored 29 goals in 65 appearances in Italy, will wear the No11 shirt after Roberto Firmino switched to No9.
“Mohamed has the perfect mix of experience and potential — this is a really exciting signing for us,” said Klopp.
“He knows the Premier League, he has pedigree in the Champions League, and he is one of the most important players for his country.
“His record in Italy has been outstanding and he possesses qualities that will enhance our team and squad.”
While Salah will be the second arrival of the summer at Anfield, following Chelsea striker Dominic Solanke, there are a number of players surplus to requirements and Liverpool have already turned down an £11m bid by Napoli for out-of-favour left-back Alberto Moreno.
Napoli are expected to return with an improved offer, but face competition from two Premier League clubs and his former club Sevilla.
Roma have been linked with Stoke’s Xherdan Shaqiri as a replacement for Salah, but Stoke chairman Peter Coates insists the club has “no plans” to sell the winger.
Shaqiri has shown flashes of the ability that prompted the Potters to pay £12m for his services in 2015, but he has also delivered inconsistent performances and suffered from numerous niggling injuries.
However, Coates is adamant the former Bayern Munich and Inter Milan man is happy at Stoke and is confident he can play a big part in the forthcoming campaign.
Asked about a possible enquiry from Roma, Coates told the Stoke Sentinel: “No, there is nothing in that. There are no plans to sell him, either. He seems very settled with us and, towards the end of the season, he came back [from injury] and did well.
“We are settled with him, too, and are hoping to see a good season from him next season.”
Meanwhile, Crystal Palace look to be closing in on the appointment of Frank de Boer as their new manager.
The Eagles are looking for their eighth boss in seven years in the wake of Sam Allardyce’s shock resignation, having kept the London club in the Premier League. Several candidates are said to have been shortlisted, including Marco Silva, who has since taken over at Watford following his departure from relegated Hull, Christophe Galtier, Mauricio Pellegrino, and Burnley boss Sean Dyche.
However, reports in the Netherlands suggest former Ajax, Barcelona, and Rangers player De Boer is in pole position to take over.