While champions Manchester City underlined their determination to hold onto the title with a come-from-behind win over Southampton and Chelsea, inspired by new signing Eden Hazard, swept to the top of the table with wins over Wigan and Reading, Sir Alex Ferguson's United are pointless thanks to their tepid display at Goodison Park.
Ferguson's side allowed Everton midfielder Marouane Fellaini to bully them into submission and United offered little in response, even when £24 million newboy Robin van Persie came on for his debut in the second half.
United can ill afford to fall six points behind City and Chelsea even at this early stage, but Giggs is confident the Everton defeat will serve as a wake-up call for his team-mates.
"It wasn't a nice feeling to lose the title last season - it never is, but it was the manner in which we lost it which was tough to take," Giggs said.
"I'm sure if we put that much effort in again and get that little bit of luck then we'll go that extra step.
"The van Persie signing sends out a big message. And it's great when a player comes out and says he wants to come to the club as well."
Ferguson could hand van Persie his first start on Saturday, while Jonny Evans should return from injury to ease United's defensive injury crisis.
If United do see off Fulham at Old Trafford, it will give City's trip to Liverpool 24 hours later even greater significance.
City will be without Argentina striker Sergio Aguero as he recovers from a knee injury suffered against Southampton and boss Roberto Mancini will hope the Argentine's absence doesn't expose the club's failure to sign a new striker.
Mancini, who wanted to buy van Persie from Arsenal, has been frustrated by City's lack of transfer success and, with Aguero out, he has the difficult task of keeping the unpredictable Carlos Tevez fully focused on leading the attack.
Liverpool manager Brendan Rodgers would dearly love to upset the odds on Sunday as he looks for his first Premier League win in charge of the Reds.
Rodgers' side crashed to a dismal 3-0 defeat at West Bromwich Albion on the opening day of the season and the after-effects of that loss could take their toll against City as they will be without suspended Denmark defender Daniel Agger following his dismissal at the Hawthorns.
In contrast, Chelsea head into their home clash against Newcastle in buoyant mood after going top of the table for the first time in 21 months.
After winning the Champions League and FA Cup double in May, Premier League success is boss Roberto Di Matteo's top target this season.
Chelsea finished a disappointing sixth place last term and Blues defender Gary Cahill, a scorer in Wednesday's 4-2 win over Reading, said: "The manager stressed before the Reading game that we haven't been top of the league for a long time and, although it's early, it's nice to be sitting there.
"It is something we needed to address this season and it has been drilled into us. It's important that we challenge this year and we have made a good start."
On Sunday, Arsenal travel to Stoke looking to lift the gloom around the north London club since van Persie's departure.
Arsene Wenger's side were held to a 0-0 draw by Sunderland last weekend, prompting fears the Gunners would suffer the same kind of dismal run that followed the departures of Cesc Fabregas and Samir Nasri 12 months ago.
But Wenger said: "What happened to us a year ago was very different to the situation we find ourselves in now and we are far better prepared this time.
"We prepared for van Persie to leave and recruited early with Lukas Podolski and Olivier Giroud and we have added to that strength with Santi Carzola.
"I believe we have a much stronger squad now than we had a year ago."
Aston Villa v Everton, Chelsea v Newcastle (1630GMT), Man Utd v Fulham, Norwich v QPR, Southampton v Wigan, Sunderland v Reading, Swansea v West Ham (1145GMT), Tottenham v West Brom
Liverpool v Man City (1500GMT), Stoke v Arsenal (1230GMT)