The Exiles are bottom of the league but have not lost since their 3-1 defeat at Stevenage on Saturday, 7 January.
After a run of seven straight defeats, Westley's side have drawn four times and won once in their last five games.
"We've taken ownership of the problem and we're fighting to put it straight," Westley told BBC Radio Wales Sport.
Westley continued: "Five unbeaten is a great initial statement from the squad and we'll look to make it six unbeaten [against Grimsby Town].
"We are very focussed with digging ourselves out of the hole we find ourselves in and push the club on to highs rather than lows.
"There's 51 points to be won and provided we keep building on what we're doing at the moment, we're confident we can win a lot of those 51 points."
The Exiles face a Grimsby side on Tuesday who lost their last game 5-0 against Crewe Alexandra, but Westley does not underestimate their midweek opponents.
"I went up and watched the game on Saturday and in fairness, it was very much a game of two halves. They just got it wrong and in the second half they tidied themselves up and they looked a decent outfit so I'm sure one heavy defeat isn't going to kill them," Westley said.
"We expect them to come full of intention and full of renewed motivation and we know that we're going to have to be as good and better than we were on Friday if we are to secure the three points."
Newport County boss Graham Westley says their recent unbeaten run is a "statement" from his side in their bid to escape relegation from League Two.