Beckford finally opened his Premier League goalscoring account by curling home a brilliant late equaliser against Bolton on Wednesday.
It ended a long wait for the 26-year-old who, after proving prolific at League One level with Leeds, initailly found the step up to the top flight difficult after his summer move from Elland Road.
Moyes said: "It's a big leap, two divisions - it is a big thing we are asking him to do.
"What I didn't want was a lot of media putting pressure on him to come into the team and score goals.
"But it wasn't the media, it was me. That is why I took responsibility.
"I put him in the team at the beginning of the season for different reasons - (Ayegbeni) Yakubu wasn't ready, we had a couple of injuries.
"I think I probably did him a bit of damage. I think I should have waited a bit longer and got him off the bench, given him 20 or 30 minutes to get used to it.
"But what he did the other night is what everyone knows he is capable of - he scored a really good goal.
"I think scoring your first goal for any club is a big point, certainly a Premier League goal, and he'll feel much better for that, I'm sure."
Despite Beckford's effort and the return to fitness of Yakubu and Louis Saha, Moyes is still concerned by his team's lack of goals.
Yakubu's winner against Stoke a fortnight ago is the only other league goal to have come from an Everton striker all season.
Saha replaced Yakubu in the side on Wednesday - the Frenchman's first start since the opening day of the season - but was unable to capitalise on a number of openings despite an otherwise lively performance.
Moyes said: "We are creating opportunities without making them into chances.
"We had opportunities against Bolton but never made the goalkeeper make any saves.
"We had opportunities to make him make saves but didn't hit the target, didn't shoot or didn't cross at the right time. A little bit of decision-making is part of it."
The Bolton result came after Everton also had to come from behind to claim a point against Blackpool last weekend.
But the failure to create and take chances aside, Moyes has been pleased with the performances of his team, who are now unbeaten in seven matches.
The Scot said: "We might have been better beating Blackpool and losing to Bolton - that would have given us three points rather than two from the two games.
"But the players showed great character. They never gave up even when we were down to 10 men and came back against Bolton.
"I have to say well done. They are doing OK, playing OK, without really finishing the games off."
The Merseysiders now face a tough test against Arsenal on Sunday but Moyes will still concentrate on developing his own team's style rather than worrying about the opposition.
He said: "I don't think Arsenal will be coming to Goodison trying to stifle Everton.
"I think they will come and do what we would do, they will play their game.
"But when you play Arsenal or any other team near the top you have to play well to get a result."
The Gunners ran out 6-1 winners at Goodison in the first game of last season but Moyes considers that a blip, overshadowed as it was by the Joleon Lescott transfer saga.
"The first game last season there were other reasons for," Moyes said.
"The games since then, we played really well at the Emirates against them and the season before at Goodison we played well."
Moyes has also given his backing to Blackpool manager Ian Holloway, who controversially made 10 changes after their creditable draw against Everton for their midweek clash with Aston Villa.
Moyes said: "Managers have to do what is right for their club and I don't have a problem with that.
"It has not got an awful lot to do with the Premier League.
"They have a good team at Blackpool. They should be praising Ian Holloway rather than in any way trying to criticise him or knock him down."