The Exiles are bottom of the table following Saturday's 2-1 defeat against Mansfield Town, and are six points adrift of safety.
"I know we've got a lot of work to do and I know we've got to win football matches," Westley said.
"But honestly I think we're capable of winning football matches."
Ryan Bird had given Newport an early lead against Mansfield but Krystian Pearce levelled for the home side.
County's Jaanai Gordon was sent off for a second yellow card before Shaquile Coulthirst's penalty secured Steve Evans' side the win.
Newport County had been unbeaten in five games before defeat at Cambridge United and although they secured a point at home to Morecambe, lost at Mansfield.
The Exiles host the side one place and three points above them in the relegation zone, Leyton Orient, on Saturday
"It doesn't matter who we win the points against, we've got to go and win some football matches," Westley added.
"We'd been on a good unbeaten run until we lost at Cambridge and reacted well in midweek to earn another hard earned point.
"I think the lads have come together really well and it's good that we've got a clear week ahead of us because one or two of the knocks can clear up.
"We've got some good footballers here and we've been in a good run of form on the whole."
Newport County manager Graham Westley is confident his side can secure the points needed to pull away from the League Two relegation zone.