A Zoltan Gera cracker just before the break, Peter Odemwingie's 64th minute penalty, and Romelu Lukaku's header sealed a famous opening day win for new Albion manager Steve Clarke, Dalglish's assistant at Liverpool last term.
It could have been worse for five-times European Cup winners Liverpool had Shane Long not missed an awful penalty on the hour after Daniel Agger's red card for a push.
Rodgers felt the decisions for the penalties were harsh.
"From being largely in control of the first half we went in 1-0 behind. A couple of decisions went against us," he told Sky Sports after handing debuts to Joe Allen and Fabio Borini.
"I've got to give the players credit, it was 3-0 but they never stopped."
Arsenal manager Arsene Wenger had almost as bad a day as Rodgers, watching his toothless side held 0-0 at home by Sunderland in the week last season's top scorer and talisman, Robin van Persie, joined rivals Manchester United.
New strikers Lukas Podolski and Olivier Giroud had decent chances but never came especially close as Arsenal supporters, fed up by high-profile departures and no trophies in recent seasons, were left fretting over another difficult campaign.
News at full time of midfielder Alex Song heading to Barcelona hardly improved the mood in north London.
Brendan Rodger's woes with Liverpool were made more galling by a thumping 5-0 win at Queens Park Rangers for Swansea City, the team he left after devising a style of play that wooed a Liverpool hierarchy keen to return to the pass-and-move groove of the 1980s.
New Spanish midfielder Michu netted the first Premier League goal of the campaign after eight minutes thanks to QPR keeper Robert Green's mistake and later grabbed another as Michael Laudrup started life as Swansea coach in thrilling fashion.
Fulham's Croatian Mladen Petric, another signing who went largely under the radar when drafted in from coach Martin Jol's former side Hamburg, enjoyed a dream debut with a brace in Fulham's 5-0 home victory over Norwich City.
West Ham United's top-flight return was marked by a 1-0 win over Aston Villa and their new boss Paul Lambert, although Kevin Nolan's 40th minute goal had a hint of offside.
New boys Reading drew 1-1 at home to Stoke City after Adam Le Fondre's late spot kick for the hosts partly spared the blushes of keeper Adam Federici, who was at fault for Michael Kightly's opener.
Newcastle United host Andre-Villas Boas and Tottenham Hotspur later on Saturday with champions Manchester City hosting Southampton on Sunday when Chelsea go to Wigan Athletic.
Manchester United and Van Persie travel to Everton on Monday.