Ben David opened the scoring for the away side with a great finish into the top corner, but Edgehill levelled through a Jimmy Beadle header.
Dan Freer restored the lead for Newlands and a second Ben David goal, which was a cracking half-volley across the Edgehill keeper and into the top corner, put the away side in the driving seat.
Freer then completed his brace to make it 4-1 with another goal which went in off the inside of the post.
Former Edgehill midfielder Paddy Mancrief is the Newlands boss this season and he was delighted with his side's performance at his old club.
"I am chuffed to bits with the win as all the lads played superbly, but man of the match was 37-year-old centre-back Jamie Gallagher, who marked Jimmy Beadle and did a brilliant job."
Itis Itis Rovers, another side under new management, continued their 100% start to the season with a 6-1 home success against Filey Town.
Sluggish defending from Town let Liam Vasey latch onto a loose pass and his deflected shot squirmed past the Filey gloveman.
Mike Barker then made it 2-0 to Jordan Lee's side and an outstanding team goal finished off by George Rose put Rovers well clear, Barker added his second before the break to make it 4-0.
Filey stepped up a gear after the break and pulled a goal back through Darren Lang, but then Liam Rackham raced forward from left-back to net a fifth for Itis Itis.
The home side capped the win when a great long-range strike from Ryan Matson flew into the net.
Man of the match for the victors was centre-back Callum Gravestock, with Danny Exley also impressing on his debut.
Ayton started life back in Division One with a hard-earned 3-2 home win against Scalby.
An own goal by the Scalby keeper and an Eric Hall effort put the hosts in front, but a Max McNiven penalty and a Zam Deans goal levelled the scores at 2-2.
The winner came thanks to a counter-attack led by man of the match Tyler Beck, Hall applying the finishing touch to secure the three points despite Scalby protesting that there was an offside.
Scalby boss Sam Medd praised all his players for a good performance in a game he felt they should have got something out of.
Seamer Sports powered into the next round of the League Cup with a 9-3 home win against Division Two side Cayton Athletic.
Seamer man of the match Sam Whitehead banged in four goals, with Benny Davis slamming in a hat-trick, including the goal of the game, a superb shot which flew in off the bar.
Danny Glendinning and Dean Craig also notched for the home side.
Luke Priestley scored a hat-trick for the visitors.