The former West Brom forward was injured in the first half of the game at the City Ground and Davies confirmed details of an injury that took the gloss off their cup win.
"Simon Cox has damaged some ligaments in his ankle, it was a freak accident and the ankle is twisted," said Davies.
"Unfortunately he has some swelling in his ankle and he will be missing for two or three weeks.
"I thought it was his knee at first, but it was not. It was his ankle, but even that is the best case scenario, in terms of the severity of the injury.
"When you start the season and you get players with different fitness levels, this kind of game is important.
"You want to win, to get into the next round of the Cup. The second thing is that you want to get minutes into players, to give them that fitness."
As for the game itself, Davies insists they are a work in progress, after goals from Radoslaw Majewski, Greg Halford and substitute Matt Derbyshire saw them advance, despite Neil Austin pulling one back for the visitors.
"We are not fluid, we are functional at the moment. We are a work in progress. We scored some good goals, but I was disappointed that we did not keep a clean sheet," he said.
"We looked a weakened side, because there are players in there who needed fitness. One or two people needed games.
"Some of the players were good, some were bad and some were indifferent, but there is no question that we have work to do. I am just happy to be in the next round."
Meanwhile, Hartlepool boss Colin Cooper suffered a frustrating evening on his return to the club he played for.
"We are trying to find the right players in the right areas, in the right combinations all over the field," he said. "Hopefully we can make progress, but we are still getting to know each other.
"I can't ask anything more from the players at the minute, but in the coming months, I will be demanding more quality from these players."