Having beaten Wolves 2-1 at home last weekend, the Toffees moved out of the bottom half up to eighth in the table. Prior to those matches Moyes' side had lost five out of six league fixtures, and the Scot is optimistic that they are now "picking up" as they have done in previous campaigns.
Moyes said: "I'm hoping that this could be the start of us moving forward."
He added: "We have been slow starters in the past, although I didn't actually think we had started slow this year.
"We lost the first game, but we got a few points on the board. Then we had a run of fixtures with a lot of the big teams in a row, but I hope this is the start of us picking up, because we normally do and when we do, we tend to stay on it and go with it."
Everton took the lead four minutes after the break when Marouane Fellaini swept in from close range, before substitute Apostolos Vellios netted a second 12 minutes from time.
Bolton had earlier had David Wheater sent off, the defender being shown a straight red card in the 20th minute for a lunge on Diniyar Bilyaletdinov after losing control of the ball, his second dismissal of the campaign.
Asked if he thought referee Michael Oliver had made the right decision regarding Wheater, Moyes said: "I did at the time and I was right in front of it.
"I don't think David Wheater is by any means that type of player, but his touch just overruns the ball and he definitely catches Bilyaletdinov.
"I'm standing five or six yards away from it and it was the correct decision. If it had been one of my players, I would have expected a red card."
Source: PA
Source: PA