The Lambs' hopes of becoming the first non-league team to knock out a top-flight side since Sutton beat Coventry in 1989 were hit inside the first five minutes when Johnny Heitinga put the hosts in front.
But Tamworth gave as good as they got after that, particularly in the latter stages of the first half, and it was not until Leighton Baines made it 2-0 from the penalty spot 11 minutes from time that victory was sewn up.
Moyes said: "Tamworth did really well, kept at it and, while it was only 1-0 there was always a chance. We missed a few chances to make it 2-0 early on, which might have put it to bed, but for the neutrals it kept the game alive.
"They threatened a bit from corners and free-kicks but, without playing well, I thought we had a level of control that was enough to probably see the game out."
Everton's lack of financial clout has made challenging for the top spots in the Premier League very difficult and a decent run in the cup would be a welcome distraction.
Moyes said: "For everybody at Everton the cup run is important, although I could say that and tomorrow we could draw Manchester City away and then it's not what you want. Once you get two or three rounds through then you can starting talking about cup runs.
"Even before today's game you're always nervous, worried about a banana skin, and I've got to praise the players because their attitude was right."
Everton needed only five minutes to take the lead, Heitinga heading in Landon Donovan's corner, and Victor Anichebe, Diniyar Bilyaletdinov and James McFadden all missed chances to double their advantage.
But Tamworth came more and more into the game as the first half wore on and striker Kyle Patterson almost scored a goal worthy of the Premier League after running from the halfway line to the edge of the area before planting his shot just wide.
The second half was a scrappy affair but the decisive moment came in the Tamworth box when Sam Habergam was ruled to have pulled down Royston Drenthe.
Substitute Baines confidently dispatched the penalty with 11 minutes left and Everton almost netted a third but Donovan's shot hit the angle of post and bar.
Everton have been hit hard by injuries recently and the one major negative for Moyes was the sight of Seamus Coleman limping off with a recurrence of a thigh problem.
The Toffees boss added: "Everybody's going to get injuries, and we don't need any more, but if we are going to get injuries it might as well be when the transfer window's open because you can attempt to do something about it."
Tamworth boss Marcus Law was proud of the display of his players but could not help but rue the circumstances in which they went behind.
He said: "There's a part of me very disappointed because we spent a lot of time on dealing with set-pieces. I've scrutinised about six DVDs of Everton, been up here for two games and that was where I felt their strengths were.
"Maybe the occasion and lack of focus in that first five minutes has gone against us. But it just shows you what small margins there are in football.
"I know these boys, I work with them every week, they're a feisty group, very mentally strong, and I knew, regardless of what was going to happen today, we would be able to compete.
"I think you can see the difference between a mid-table Blue Square Premier side and an elite Premier League club was a set-piece and a questionable penalty."
Of the spot-kick, Law added: "I think it was too easy to give. The player's got himself close, our defender couldn't really get out of the way. He is the wrong side of the ball so I think the ref's sucked into it really."