Rhys Day levelled the scores when he rose highest to power in Ryan Williams' left-wing corner from eight yards before Exeter keeper Paul Jones was sent off three minutes later to leave Paul Tisdale's side in real trouble.
A long ball over the top sent John Grant through on goal but he was cleaned out in full flow by Jones some 20 yards from goal.
Striker Richard Logan was sacrificed for sub keeper Martin Rice and Dixon gave the home side the lead after just 13 seconds of the second half.
Dixon reacted quickly to cleverly flick the ball over Rice from the edge of the area and increased the lead shortly after the hour with an even better finish.
Steve Tully lined up a free-kick wide on the right but squared it to Dixon, who hammered home from the edge of the area to seemingly wrap up all three points.
But Exeter were right back in it when Daniel Seaborne flicked in Matthew Gill's 63rd minute corner from the edge of the six-yard box and sensed a draw when Aldershot were reduced to 10 men with 10 minutes remaining.
Ryan Scott launched into a bad tackle on Steve Tully and was instantly dismissed by referee Gary Lewis but Exeter were unable to capitalise.