Quickfire goals from Diogo Jota and Darwin Nunez before the break and Mohamed Salah's second-half effort helped Jurgen Klopp's side maintain their 100 per cent home record in the Premier League this season on Sunday (Monday AEDT).
Jota held aloft Diaz's number seven shirt after scoring, with the Colombian not featuring in Liverpool's squad following reports his parents had been kidnapped in his homeland.
Liverpool's latest victory - their eighth in a row in all competitions at Anfield - was their seventh from 10 league games in this campaign and lifted them back to within three points of leaders Tottenham.
Forest rarely threatened and, although they hit the woodwork through Anthony Elanga's late volley, they were forced to defend in numbers as their winless league run was stretched to six matches.
A minute's applause was held before kick-off in memory of former England and Manchester United great Sir Bobby Charlton and ex-Everton chairman Bill Kenwright, who both recently died.
Two goals in four first-half minutes gave a fair reflection of Liverpool's control.
Nunez's fierce first-time shot was parried by Turner into the path of Jota, who turned home the rebound from eight yards.
In celebration, Jota ran to manager Jurgen Klopp, who handed him Diaz's shirt to show to the Liverpool fans.
Nunez doubled Liverpool's lead in the 35th minute with an emphatic near-post finish after Dominik Szoboszlai's cut-back.
Salah capitalised on Turner's misjudgement to put Liverpool 3-0 up in the 77th minute.
Turner was caught out by the bounce of Szoboszlai's long diagonal punt up field and Salah raced clear to sidefoot home his eighth league goal of the season.
Earlier, Dominic Calvert-Lewin's landmark second-half strike secured a 1-0 win for Everton at West Ham.
The striker scored six minutes after halftime at the London Stadium on Sunday (Monday AEDT) as Everton won for only the third time in the league this season.
For Calvert-Lewin, who has had to contend with a slew of injuries, it was his 50th league goal for the club as he joined Romelu Lukaku, Duncan Ferguson and Tim Cahill in reaching the half-century mark.
Receiving the ball in the box, he turned smartly and steered a shot low into the corner.
Aston Villa maintained their strong start to the season with a dominating 3-1 victory over newcomers Luton, their fifth win in six league games.
John McGinn, Moussa Diaby and an own goal from Luton captain Tom Lockyer made it a comfortable afternoon's work for Villa, before a consolation 83rd minute own goal from the home goalie Emiliano Martinez.
In the day's other game, Fulham escaped with a 1-1 draw at Brighton, who dominated the first half to go in one-up through an Evan Ferguson goal but were then pegged back by a stunning Joao Palhinha equaliser in the 65th minute.
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