Both teams finished the tie with 10 men.
However, goals from Jamie Ford and Ryan Pennery gave Moneys the verdict.
‘It was not the best performance from us but we defended well when we had to,’ said boss Carter.
‘VCD were a young, quick, fit side and we had to work hard for it.
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‘Though it sounds silly we scored too early in both halves.
‘Having said that we took both our goals well.
‘The most important thing in the Cup is making sure you are in the hat for the next round and we achieved that.’
Moneys made a dream start – taking the lead after just three minutes.
Steve Hutchings, in his first start of the season after a five-game suspension, made his presence felt immediately.
His header from a corner was pushed out by the VCD keeper to the unmarked Ford, who nodded the ball in at the far post.
The hosts levelled the scores within two minutes.
‘We didn’t clear a corner and when the ball came back in we didn’t pick their runner up,’ added Carter.
‘For the rest of the half the two teams cancelled each other out.’
Moneys were quick out of the blocks again at the start of the second half.
Lewis Fennemore won the ball in midfield and fed Conor Bailey down the left.
Bailey delivered a deep cross and Pennery headed in at the back post.
‘After taking the lead again we sat off them a bit and had to defend,’ said Carter.
‘We did come under the cosh a bit but still had the better chances.’
Sam Pearce and Hutchings were denied by the crossbar before both teams were reduced to 10 men on 70 minutes.
‘Connor Hore didn’t react very well after their player stamped on his thigh and both got sent off,’ said Carter.
‘Hopefully now we can get a nice home tie in the draw for the next round.’