Westley lasted five months and became County's second boss sacked this term.
The Exiles have confirmed Mike Flynn as caretaker boss until the summer.
"We engaged and consulted a number of folk, including some of the players and given the overwhelming view of fans coupled with the players' feedback, it was obvious the environment needed a change," Foxall told BBC Wales Sport.
"We hope Michael is able to put the passion that was not there against Leyton Orient on Saturday back into the club," he continued.
"You don't get stability by changing the manager regularly.
"But looking at the run of results and more so the Leyton Orient game, the manner of the defeat, we as a board tried to consider all factors.
"The players have smiles on their faces and I do think there has been an uplift in their mood."
Foxall says he understands criticism of the board for allowing Westley to sign 13 players in January only to sack him six weeks later, but says they felt they had to back him until Newport were 11 points adrift at the bottom of League Two and in danger of relegation.
"We had a view as a board that everything Graham had he inherited and we said we would support him in changing the squad," he said.
"Hindsight is a great thing, but we felt we've always made the right decision for the club at that time.
"We have always had the view we would support the manager 100%.
"We still believe it was the right thing to give him the window we did, even though it has not worked out how we all would have wanted."
Foxall says the hope is that Flynn, a former player and local man, will be able to galvanise the County supporters.
"Michael is a local boy, born and bred and loves the club," he said.
"We are hoping as a board that his appointment will rally the fans, they have not had much to shout about this season.
"I've noticed Michael has matured greatly over the time I've been involved with the club.
"Michael was an obvious choice."
Foxall said the club would have to make some "difficult' and 'challenging" decisions if they lose their Football League status, but insisted the club have "the right infrastructure in place to challenge next season."
Newport chairman Gavin Foxall says the board consulted the players before opting to sack manager Graham Westley.