Zac Hansen's double put Newlands in control by the half-time interval, but the hosts struck two late goals and a penalty shoot-out looked to be on the cards.
Newlands refused to buckle though and Brad Marshall restored Newlands' lead after seizing upon a rebounded shot from Hansen, and the latter completed his hat-trick to secure a spot in the next round for Dan Sheader's team.
Mark Wade impressed in central midfield for Newlands.
Angel soared into the next round with a 4-0 home win against Kirklington, Danny Collins scoring all four for the hosts.
Lloyd Henderson was man of the match in an all-round fine performance from Angel.
Eastfield United slumped to a 10-2 loss at home to Coulby Newham & Nunthorpe.
Luke Kirby and Michael Putley netted for the hosts, the latter from the penalty spot, with Adam Martin named as the man of the match for United.
In the Sunday League, Valley maintained their 100% record with a 5-0 win against Saints.
Sam Westmoreland, Rich Tolliday, Harry Ward, Danny Appleby and Liam Wainwright notched for Valley, with right-back Brad Parkin their man of the match.
Trafalgar kicked off their league campaign with a 13-0 win at Castle Tavern, Liam Salt and Issy Macauley leading the way with a hat-trick apiece.
Player-manager Mikey Barker and Jamie Patterson also scored two apiece, with Danny Glendinning, Danny Price and man of the match Ted Edwards also on target.
Tavern's man of the match was keeper Connor Ramsden.