The former Danish Under-21 international, who joined Albion earlier this summer from Ipswich in a £520,000 deal, opened the scoring with his first goal for the club.
Gaardsoe struck after 11 minutes with Rob Hulse putting the result beyond doubt with two goals to take his tally for the season to six.
To complete a miserable day for Ipswich, Town central defender Drissa Diallo headed into his own net four minutes from time to complete the rout as Joe Royle's side crashed to the bottom of the table.
Ipswich, who have yet to win a game in the league this season, crashed to their third successive away defeat despite a second-half strike from Richard Naylor - the first goal on their travels this season.
Disappointed Ipswich boss Joe Royle, who has seen Town make their worst start to a season for almost 30 years, admitted: "I can't be happy with that, our defending was abysmal.
"There were two free headers, a goalkeeping error and an own goal that shows how shoddy we were."
Gaardsoe rose above the static Ipswich defence to fire Albion in front when he headed home a corner from Jason Koumas.
Ipswich were still reeling from that setback when Hulse struck from close range.
Hulse was fouled on the edge of the area by Chris Bart-Williams and from the resulting free-kick, Ipswich keeper Kelvin Davis failed to hold a shot from Koumas and Hulse rammed home the loose ball from close range.
Town's hopes of a comeback were then dashed 11 minutes into the second-half when Hulse grabbed his second with a glancing header from another Koumas corner.
But, to their credit, Ipswich kept battling and their persistence was rewarded after 63 minutes when Naylor headed home a corner from Alan Mahon.
But Albion had the last word when Diallo steered a cross from Albion wing-back Neil Clement beyond the stranded Davis and into an empty net.