Substitute Johnny Heitinga drilled home an injury-time winner after Steven Pienaar's first-half opener had been cancelled out by Marvin Sordell for the npower Championship strugglers.
Moyes made his intentions clear by naming his strongest possible side for what was his 500th game in charge of the Toffees, but conceded circumstances may affect his bid for his first piece of silverware.
Moyes said: "It is good to put my best teams out but that might run aground later on if we are running out of legs or we get injured.
"There were times I was looking over my shoulder and I didn't have a lot of options."
Moyes said striker Kevin Mirallas sustained a recurrence of a hamstring injury which forced him to be substituted inside the first 20 minutes, but indicated the injury might not be as bad as first feared.
Moyes said: "Kevin has had hamstring problems so we have got to hope it is not too bad, but I don't think it is as bad as we first thought it could be.
"You could see how well we played in the opening 15 minutes and he is a really important player for us. He helped string a lot of things together and we missed him after that - we never really got flowing.
"I thought in the second half we had most of the play without being that creative, but we looked the more likely to score.
"It was great to get the job done. I've had many years of not getting through in ties quite similar to this."
- Sports & Recreation
- Marvin Sordell
- Steven Pienaar
- Kevin Mirallas