Kyle Graham-Luke added another for a 3-1 advantage in the 57th minute, but City were just not able to put the match into deep freeze.
After experianced head Jason Fergusson withdrew with illness that morning, Josh Jonas was thrust into the pressure cooker and made a couple of hard challenges outside the box - getting two yellow cards - with Hastings converting on the second offence for 3-2.
Jordan Joblin-Hall got one back to re-establish the advantage 4-2, and Bell would later add his third goal.
But Hastings struck again and then in a horror period for Francis, he had to watch as the Rangers scored three straight goals from deflections over the keeper in almost identical circumstances - the last being a very well taken strike.
"Trying to shut a game down with ten men is difficult," he said.
With four minutes of added time available, City desperately tried to force an equaliser as Graham-Luke made a hard drive at the goal but could not connect, then a City header attempt went just over the cross bar.
"You could see it just wasn't to be," said Francis.
"We'll learn from that experiance, and it's probably a good thing it was a Cup fixture and not next week with Hokowhitu."
Adam Woon and Harry Borschmann both scored doubles for Hastings, with Dale Simpson and Benjamin Robertson getting the other goals.
Graham-Luke was named MVP for his hard work in the middle of the park.
As well as Jonas being on automatic stand down, City may not have the services of Eli Fleming and Adrian Ngaau this weekend, due to knee injuries.
Meanwhile, an understrength GVCA Wanganui Athletic Reserves also went out in the first round of the Cup, beaten 4-2 by the Gisborne Thistle.
With a one-player bench, the Horizons Premiership squad could not rein in their Lotto Federation League opposition.
Athletic's Jai Stephens was a goal scorer, while Paul Goodlett got a double for the Thistle, with the other goal scorers being David Ure and Timothy Hofman.