Dover rise to third while Wrexham are now six points off the top five after Ricky Modeste's second half winner proved decisive in a tense contest.
Modeste fired home off the post after Sam Magri's effort in a game low on quality chances.
James Gray hit the crossbar as Wrexham pushed for an equaliser, while Simon Heslop also came close.
Wrexham manager Gary Mills told BBC Radio Wales: "It's a disappointing day for us and we never really got going. I've said to the players that there's no point in us looking at that now, the final whistle's gone.
"We've got four games left . It's tougher but still doable and we've got to win on Tuesday at Guiseley.
"If we win on Tuesday then we got to Tranmere to beat them to go above them and that's as simple as it is. We battle on."

Denmark international Fischer, 21, has been with the Amsterdam club since 2011 and scored 24 goals in 79 league games.
The forward visited Boro, who were promoted to the Premier League this season, for a medical and agreed terms.
"It's the greatest league in the world and I feel really ready," Fischer told Danish TV station TV2.
Eight-times capped Fischer has agreed a three-year deal.
Fischer emerged as a key player at Ajax under recently departed coach Frank De Boer, alongside compatriots Lasse Schone and Christian Eriksen - the latter whom he will play against in next season's top flight when Boro face Tottenham Hotspur.
Dover maintained their National League promotion push at the expense of Wrexham at the Racecourse Stadium.
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Ajax winger Viktor Fischer will join Middlesbrough for around £3.8m at the start of the international transfer window next month, BBC Tees reports.