"It's a big catch-up for us now," said Moyes, whose team dropped to ninth place in the stuttering defence of its title 12 points behind Arsenal after 14 games.
Tottenham ruined Rene Meulensteen's first match in charge of Fulham by winning 2-1 at Craven Cottage, giving Spurs manager Andre Villas-Boas a first win in five games and some much-needed breathing space.
Swansea ended Newcastle's four-match winning run with a 3-0 victory and Aston Villa upset Southampton 3-2 to take the number of goals in the night's nine matches to 32. Stoke was held 0-0 by Cardiff.
Suarez became the first player to score three hat tricks against the same team in the Premier League, beginning his virtuoso performance in the 15th minute with a dipping volley from long range.
An instinctive second was followed by a classy third, when he turned this way and that before smashing a rising shot into the corner from the edge of the area. He finished off his one-man show with a curling free kick and added an assist for good measure, setting up Raheem Sterling for Liverpool's fifth.
Despite missing Liverpool's first five games of the campaign to complete a ban for biting carried over from last season, Suarez is now the Premier League's top scorer with 13 goals in nine matches.
"I wish Suarez would just leave us alone!! Big bully!" Norwich midfielder Anthony Pilkington tweeted.
Arsenal may not have racked up the goals but the win was just as routine against a team that shocked Liverpool three days earlier, keeping Arsene Wenger's side out in front.
Bendtner was handed his first Premier League start since March 2011, with Olivier Giroud given a rest, and the forgotten man responded by heading home the opener after just 90 seconds.
"When he is physically sharp, he can compete with anybody," Wenger said of the Denmark striker, who has been loaned out to Sunderland and Juventus in recent seasons. "When he has that commitment and attitude, his quality shows."
Mesut Ozil stroked home a pass from Aaron Ramsey to make it 2-0 in the 47th and secure Arsenal's 11th win of the campaign.
United struck the goal frame twice, through Wayne Rooney and Danny Welbeck, but Everton overran the hosts in midfield and deservedly prevailed thanks to Oviedo's close-range finish.
"We will need to play some great football and win an awful lot of games in the run-in if we are going to be in there and around it come the end of the season," said Moyes.
• West B Albion 2 Manchester City 3
• Fulham 1 Tottenham Hotspur 2
• Stoke City 0 Cardiff City 0
• Liverpool 5 Norwich City 1
• Arsenal 2 Hull City 0
• Southampton 2 Aston Villa 3
• Sunderland 3 Chelsea 4
• Manchester United 0 Everton 1
• Swansea 3 Newcastle United 0
- AP