Salisbury conceded a late Euan Williams goal in additional time after half-time substitute Nathan Odokonyero opened the scoring within two minutes.
Before the heavens opened just before kick-off, the Whites looked to extend their six-game unbeaten run at home in all competitions against a Dover side sitting just below the playoff places.
Most spectators at the Raymond McEnhill Stadium thought both sides were battling the conditions over each other to begin with, but Brian Dutton's side got off to a flying start, as some direct football in between the lines led to the first chance of the game for winger Robbie Gallagher after Ollie Morgan whipped in a low cross in the third minute, but it was blocked before it could ripple the net.
Despite the hosts having the lion's share of possession and chances, the visitors had a handful of counter-attacks of their own and came close to goal on a couple of occasions.
Salisbury manager Brian Dutton said he "doesn't want to be too critical" of the performance because of the Whites' fine form, but admitted it was "lacklustre" for the standards set this season.
Midfielder Caine Bradbury had the best chance of the half for either side in the opposition box, but it flew over the bar just before half-time.
A change in personnel was needed for the home side, and it came up front as top goalscorer Nathan Odokonyero entered the fray for Gallagher.
Within one minute, the ever-energetic midfielder Mo Dabre won a penalty in his second league start of the campaign.
Odokonyero stepped up after just two minutes on the pitch, then calmly slotted his spot-kick into the bottom left-hand corner, giving the keeper the eyes and sending him the wrong way.
With the advantage, the visitors made the rest of the afternoon difficult for the home side, pinning them in their half on more than one occasion.
The pressure paid off, and Dover got their equaliser after a run down the left-flank from another substitute, Sak Hassan, was cut back into space for Euan Williams to finish and snatch a point at the death.
Salisbury remain unbeaten in eleven matches in all competitions and stay in the playoff positions after 12 league matches.
They now travel to Walton & Hersham tomorrow evening (27th October) for their next league fixture. Full commentary on Salisbury Radio Sport from 7:35 pm for a 7:45 pm kick-off.

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