After examining the Umbraco source code, I was able to achieve this in front end and backend code.

In front end I created a jQuery ajax call to the following endpoint: '/umbraco/backoffice/UmbracoApi/Authentication/PostLogin'. I also passed in username and password details as JSON object.

In back end I used the following code from a UmbracoApiController:

private async Task<SignInStatus> LoginAsync(LoginModel loginModel)
    {
        // Get http context
        var http = this.TryGetHttpContext();
        // Get owin context
        var owinContext = TryGetOwinContext().Result;
        // Get backoffice sign in manager
        var signInManager = TryGetOwinContext().Result.GetBackOfficeSignInManager();
        // Call sign in method
        var result = await signInManager.PasswordSignInAsync(loginModel.Username, loginModel.Password, true, true);

        return result;
    }