Daish's side were relegated from the Blue Square Premier last season
Ebbsfleet United moved up to second in Blue Square Bet South with a 4-2 win at Hampton and Richmond.
Liam Daish's side are now six points behind leaders Braintree, who suffered a surprise 1-0 defeat by Dartford.
The Darts got their fourth straight win despite having Charlie Sheringham sent off after 20 minutes, Elliot Bradbrook scoring the second-half winner.
Dover Athletic recovered from Saturday's defeat by Ebbsfleet with a 2-0 win at promotion rivals Bromley.
"It's a massive win for us, especially after the disappointment of Saturday at home," Dover boss Martin Hayes told BBC Radio Kent.
"I can't work it out at the moment - we do play well away from home and we just can't take it to Dover and we disappoint a lot of home fans.
"It's a big three points - it puts us hopefully among the chasing pack going into the Braintree game.
"It's another big team we've had to deal with, we've got a string of back-to-back big sides and I'm very pleased for the lads, they responded well."
Bromley's new boss Hayden Bird added: "I'm not so much disappointed with the performance, I thought there were two good sides out.
"You can see why Dover have been so successful on their travels and I think for our part, small detail has lost us the game.
"We had opportunities, their first two opportunities they scored from and in the last third there's no doubt about it - they're a very good side."
The defeat for Bromley means they slip to third in the table, while the Whites remain in play-off contention in seventh.
Meantime, Ebbsfleet manager Liam Daish said that it is far too early to think about overtaking Braintree at the top, even though his side have now climbed to their highest position.
"There's so much football to be played and I think there's going to be a lot of twists and turns," he said.
"We've just got to concentrate, we've got two big games now - Farnborough and Eastleigh.
"I think they take care of themselves sometimes, it's these games that really need to be put to bed."
RESULTS - Tuesday, 2 February
Blue Square Bet South Bromley 0-2 Dover Dartford 1-0 Braintree Town Hampton/Richmond 2-4 Ebbsfleet United
Ryman Premier Division Hastings United 2-2 Folkestone Invicta Hornchurch 2-1 Margate Maidstone United 0-3 Tonbridge Angels
Ryman Division One South Chatham Town 3-1 Eastbourne Town Sittingbourne 1-1 Faversham Town
Kent Senior Cup Whitstable Town 3-0 Welling United
Kent League Cup Quarter-Final Beckenham Town 0-2 Corinthian (Corinthian win 3-2 on agg) Hythe Town 2-1 Sevenoaks Town (Hythe win 4-3 on agg)
Wednesday, 2 February
Kent League Cup Quarter-Final Herne Bay v Norton Sports
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