In your installer class of the service itself, use the event AfterInstall to start the service. After setup has finished, the service will start.

public ProjectInstaller()
        {
            InitializeComponent();

            AfterInstall += new InstallEventHandler(ProjectInstaller_AfterInstall);
        }

        void ProjectInstaller_AfterInstall(object sender, InstallEventArgs e)
        {
            System.ServiceProcess.ServiceController sc = new System.ServiceProcess.ServiceController(serviceInstaller.ServiceName);
            sc.Start();
        }

Also have a look at this Source.

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