The hosts opened the scoring when 36-year-old Andy Cole marked his debut by firing in from an acute angle.
But seven minutes before half-time Jackson beat Burnley goalkeeper Brian Jensen after David Unsworth's foul on Scott Vernon.
Colchester's Dean Gerken saved expertly from Robbie Blake's 25-yard free-kick as the visitors successfully held out.
Colchester boss Geraint Williams:
"We were disappointed to go in level at half-time as the best three chances in the first half were ours. In the first half we did enough to win the game. "Every game is vital and all the players are fighting to stay in the Championship.
"We are in a battle but at the moment everyone is rolling their sleeves up."
Burnley: Jensen, Alexander, Varga, Unsworth, Harley, Elliott, James O'Connor (Randall 73), Gudjonsson (Caldwell 52), Lafferty, Cole, Blake (Akinbiyi 61).
Subs Not Used: Kiraly, Spicer.
Goals: Cole 23.
Colchester: Gerken, Ifil, Coyne, Virgo, White, McLeod (Guttridge 75), Izzet, Duguid, Jackson, Lisbie (Vernon 26), Platt.
Subs Not Used: Cousins, Watson, Sheringham.
Goals: Jackson 38.
Att: 15,376
Ref: Steve Bratt (W Midlands).
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