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Klopp believes goals will come for 'exceptional' Nunez
Salah gave Liverpool a strong start when he converted Andrew Robertson's delivery from close range – making the left-back the defender with the most Premier League assists in history.
Egypt star Salah then teed up Virgil van Dijk to steer home a deflected second, though the Reds came under pressure after Ollie Watkins halved the arrears with just over half an hour remaining.
Jurgen Klopp's side made the points safe late on as 18-year-old substitute Bajcetic rounded Robin Olsen to score his first Premier League goal, moving Liverpool within five points of the top four.
Liverpool needed just five minutes to take the lead, with Salah on hand to tap home Robertson's cross.
50/50 - Mohamed Salah (125 goals & 50 assists) has become just the second player to both score 50+ goals and assist 50+ goals for Liverpool in the Premier League after Steven Gerrard (120 goals & 92 assists). Greats. pic.twitter.com/bYi8nrRBfA— OptaJoe (@OptaJoe) December 26, 2022
That represented Robertson's 54th Premier League assist – one more than former Everton and Wigan Athletic left-back Baines managed before his retirement in 2020.
Trent Alexander-Arnold – who sits third on that list with 45 assists – showcased his own vision for Liverpool's opener with a sublime pass to release his fellow full-back Robertson.
A lively first half saw both goalkeepers forced into action, though Watkins should have done better when heading at Alisson before Robin Olsen was tested by Darwin Nunez's volley.
After a series of narrow escapes, Villa failed to prevent Liverpool from striking again 37 minutes in, as van Dijk's left-footed volley struck Ezri Konsa before finding the bottom-right corner.
The offside flag denied Watkins a goal soon after the restart but there was to be no reprieve for Liverpool when he nodded Douglas Luiz's cross home after 59 minutes.
Liverpool then went close to a third through Nunez before Bajcetic stepped off the bench to calm any nerves, latching onto a loose ball to round Olsen and lash a left-footed finish home.
2 - Liverpool’s Stefan Bajcetic (18y 65d) has become the second youngest Spanish scorer in the Premier League after Cesc Fàbregas for Arsenal against Blackburn in August 2004 (17y 113d). Entrance. pic.twitter.com/dWouSnHDWa— OptaJoe (@OptaJoe) December 26, 2022
The Reds host Leicester City in their final Premier League game of the year while Villa's 2022 is over –next in action at Tottenham on New Year's Day.
In other Premier League results on the first day of action after Qatar 2022, Wolves came from behind to win 2-1 at Everton courtesy of a stoppage-time goal from substitute Rayan Ait-Nouri.
The points lifted it off the foot of the table with Southampton now bottom after the Saints lost 3-1 to Brighton & Hove Albion.
Fulham moved into the top half with a convincing 3-0 win at mid-table Crystal Palace, which played more than half an hour with nine men after Tyrick Mitchell was shown a straight red in the 34th minute and James Tomkins received a second yellow and his marching order in the 57th.