Kozachik had more than 57 percent of the vote, while Miranda Schubert had 27 percent and Andres Portela had 15 percent in the Democratic primary.
"My staff and I work hard at building relationships," Kozachik. "That is important in our community. My team and I are grateful to the community for their support."
Portela said the result wasn't what he hoped for, "but I'm forever grateful for those who supported me."
In Ward 3, Dahl had 60 percent of the vote, while Juan Padres had 40 percent.
Dahl, who focused his campaign on responding to climate change, said he was pleased with the results.
"I think one thing it shows is that Tucson voters understand that climate change is not just an issue that will affect the environment and our water supply, but it will impact people, especially those who can least afford to respond," Dahl said.
Padres congratulated Dahl on his victory, saying his opponent "ran a solid campaign."
There's no incumbent in the Ward 3 race as Councilman Paul Durham stepped down earlier this year and Karin Uhlich, who was appointed to replace Durham, did not run for the seat.
Kozachik will face independent candidate Val Romero in the November general as he seeks a fourth term. Dahl is set to face Republican Alan Harwell Jr., who got enough write-in votes to land a spot on the general election ballot, and independent candidate Lucy LiBosha in November.
In Ward 5, Councilman Richard Fimbres faced no primary opposition and will face no opponent in the November general election.