The Welsh club moved into fourth place and the play-off positions courtesy of efforts in either half from Harry Worley and Conor Washington.
And Newport boss Edinburgh has been delighted with his newly promoted side's dominance on their own turf this season where they have won their last five outings.
Edinburgh said: "The win is extremely pleasing because we came up against a good side. We took our chances when they came along so I'm delighted with the result and the performance.
"It's always good to be in the play-offs. It's a good place to be and I'd rather be there than down at the bottom. When you come into a new league you want to be competitive and we've continued the momentum from last year. We like the success and we'd like to continue that.
"We will look at the results on Saturday and hopefully we will stay in our position. We want to win every game we play here. It's a really good place to play right now."
The Welsh club did not look back after taking a sixth minute lead when Robbie Willmott delivered a right-wing corner and centre-back Worley rose highest to plant his header past Scott Flinders.
Hartlepool almost drew level as Deji Oshilaja allowed the ball to run to Jack Compton, whose shot was blocked by Ismail Yakubu and James Poole's follow-up effort was routinely saved by Newport goalkeeper Lenny Pidgeley.
Moments later, the former Chelsea stopper produced a flying save to deny dangerman Poole.
Washington sealed the win in the 63rd minute with his seventh goal of the season from close range after neat work by Christian Jolley down the right flank, although Hartlepool's Andy Monkhouse hit the post in injury-time.
Hartlepool manager Colin Cooper said: "We've had a long chat and the players have been very honest. They felt they have fallen below the standards they have set for themselves.
"It's a fair assessment.
"I can't be too critical. Playing against Newport wasn't about standards, it was about rolling our sleeves up. Newport tested us and we didn't really stand up to the test.
"They are effective. I still prefer to do things my way, but Justin has got his side well-drilled, they know their jobs and are tough to beat."
- Sports & Recreation
- Harry Worley