New Year's Eve saw big guns Manchester United and Chelsea suffer shock defeats at home to Blackburn and Aston Villa respectively, before league leaders Manchester City were beaten by a last-gasp goal at Sunderland 24 hours later.
Everton, meanwhile, extended their unbeaten run to four games with a 1-0 win at West Brom and the odds look good for them continuing the sequence with another triumph on Wednesday night when they host Bolton.
Wanderers are bottom of the table and lost 2-0 at home to Everton in November, but with the top flight seemingly as unpredictable as ever, Moyes is not about to underestimate his team's task at Goodison Park.
"The Premier League is back to the topsy-turvy way it was last year when you didn't know who was winning what," Moyes said.
"Just over the last couple of weeks we have started to see that again. There is no guarantee when you play any games who is going to win or lose at the moment, so you need to go about your job correctly.
"We got a good three points (at the Reebok Stadium) and it would be great if we could do it again, but there is nothing guaranteed and I wouldn't take anything for granted.
"Bolton are difficult opposition - they have their own fight at this moment in time while we are trying to push up the league."
Moyes will be able to call on Landon Donovan on Wednesday for the first time since it was announced in December that the United States international would be rejoining the Toffees on a two-month loan deal from the Los Angeles Galaxy.
Donovan proved a success in a stint of the same duration with Everton in 2010 and says he feels "more prepared" this time, although Moyes has stressed it is only when he takes part in matches that the 29-year-old can properly be assessed.
Asked if he thought Donovan had "hit the ground running" in training, Moyes said: "He looks okay, I have to say, but how can you tell?
"Even though we have seen some football, we have not seen any games yet.
"But fitness-wise he is not someone who carries a lot of weight. He is quite lightweight and he looks okay.
"He came before and did well for us - scored a couple of goals, played well and showed some qualities.
"We are hoping over his time here he will do that again, but we have to remember it's only a short loan.
"He has settled in very quickly, so we look forward to getting him in the game and seeing how he does."
Everton have announced that players will wear black armbands on Wednesday night as a mark of respect to Gary Ablett, and that there will be a period of silence ahead of the FA Cup match with Tamworth at Goodison Park on Saturday.
It was confirmed on Monday that former Toffees defender and coach Ablett, who also played for Liverpool, had died at the age of 46 after a 16-month fight against non-Hodgkin's lymphoma.
Speaking about Ablett on Tuesday, Moyes said: "It was a really sad moment when I heard about Gary.
"He has been coming to our training ground quite regularly since he became ill. I went to see him in hospital when he first got diagnosed and since then I've kept a real close watch on how he is going.
"I think there were good signs at times, but then we heard bad news at different times that he wasn't so well.
"He was in with his boys not too long ago and he came and watched a bit of training - we tried to keep him involved as much as we could. But I think in his own mind, Gary probably knew himself - I'm sure he did.
"All I can say is that my thoughts are with his wife and family, and we will miss him greatly here.
"It is a great thing to be able to say that he played for both sides.
"They are both great clubs and I'm sure Gary was very proud of that. It just shows that you can do that and be respected from both sides of the city."