Ignacio Mieres and Tom Homer traded three points in the opening 15 minutes of the first half, before the Argentinian added a second to slip in front.
James Hanks added a late try to the home team's total, and set up Mieres to add another couple to his tally ahead of the half-time break.
Homer continued to produce for the visitors in the second half with the team's first five points of the afternoon. He then kicked three more to reduce the deficit to two points.
However, wing Gonzalo Camacho crossed over to extend Exeter's lead with less than 10 minutes to go, allowing Rob Baxter's side to hold on and win.