It had been hoped the pair would make the gesture in a show of unity, as Liverpool played their first game at Anfield since the Hillsborough Independent Panel absolved Liverpool supporters of any blame in their findings on the 1989 stadium disaster, which took the lives of 96 Reds fans.
Liverpool forward Suarez refused to shake United defender Evra's hand in the league meeting between the sides at Old Trafford in February, having previously been found guilty of racially abusing Evra in the corresponding fixture at Anfield last season.
Sunday's handshakes were part of a well coordinated series of pre-match events aimed at ensuring the first Liverpool home game since that damning judgement on the Hillsborough tragedy was delivered passed off peacefully.
Both teams emerged wearing tracksuits bearing No. 96, while United legend Sir Bobby Charlton presented flowers to Liverpool counterpart Ian Rush before respective skippers Steven Gerrard and Ryan Giggs released 96 red balloons.
A mosaic was also displayed around three sides of the ground during the traditional playing of "You'll Never Walk Alone."
Although fears had been expressed about the behaviour of United's fans ahead of the game, early signs were encouraging.
While some sang anti-Liverpool songs on their way into the stadium, none made any reference to Hillsborough.
In addition, their supporters applauded when a tribute to the Hillsborough families was read out prior to kickoff.
Once the game started both sets of fans traded insults, although a refrain of "Where's your famous Munich song" from the United fans drew boos.
Earlier, Charlton had been greeted with warm applause on his arrival at the stadium, while fellow 1968 European Cup winner Paddy Crerand chatted with home supporters.
And, while Sir Alex Ferguson, Robin van Persie and Evra, who had been overlooked for the captaincy in the absence of Nemanja Vidic, were booed as they made their way off the team coach, the reaction was half-hearted.