Jackson (1-0), pitching on his 20th birthday, bested lefthander Randy Johnson, allowing one run and just four hits. The righthander did not walk a batter and struck out four.
Eric Gagne converted his 58th straight save opportunity.
Arizona took an early lead in the second inning on singles by Raul Mondesi and Shea Hillenbrand, and a sacrifice fly by rookie Robby Hammock.
After that, Jackson retired 13 of the final 15 hitters he faced. Paul Shuey pitched two hitless innings before Gagne retired all three batters he faced for his 50th save of the season.
The Dodgers, who won their fifth in a row, remained two games behind Florida and Philadelphia in the National League wild card race. They scored three times off Johnson in the third.
Johnson (4-8) lasted eight innings, allowed four runs, three earned, and eight hits.
Los Angeles has won 10 of its last 12.