The American stopper, 34, sustained the fractures in the 85th minute of the Toffees' FA Cup fifth-round replay victory over Oldham Athletic, when he landed awkwardly following a foul by striker Robbie Simpson.
Despite his injuries, Howard stayed on the pitch for the remainder of the match, making several saves as Everton held out for the 3-1 win.
That result set up Saturday's quarter-final against Wigan Athletic, which Howard will miss.
"Tim is out against Wigan," said Moyes. "He has broken two bones in his back. We don't think he will be out for the season.
"I have let him go away for a few days. It might be that, if everything goes well, he misses next week's game [against Manchester City] as well and then, because of the international break, he might be close after the rest.
"He has two fractures, but they are not bones that are supporting anything. It is off the spine. They are not weight-bearing. It just needs a bit of healing time."
Howard's absence means that understudy Jan Mucha will make his FA Cup debut at Goodison Park.