Connor Jennings scored twice for the Silkmen, who knocked Swindon out of the competition last season, with Scott Boden and Peter Winn also on target.
Tamworth also beat league opposition as they saw off Cheltenham 1-0, the only goal coming in the 20th minute from former Everton striker Nick Chadwick.
There was nearly a massive upset at Priestfield where Gillingham needed a goal from Bradley Dack deep into injury-time to secure a 1-1 draw with Skrill North side Brackley Town.
The Saints appeared to be on their way to the second round for the first time when home keeper Joe Martin put through his own net 21 minutes from time.
It was a similar story at the Kassam Stadium where Gateshead led Oxford United 2-0 but were pegged back to 2-2 late on when Danny Rose headed the U's level and earned them a replay.
Braintree Town will replay with Newport after a 1-1 draw and Boreham Wood also secured a replay after drawing 0-0 with Carlisle.
But there was no dream for Daventry, the lowest ranked team left in the competition, as they were beaten 2-0 by Chesterfield, while St Albans City were the day's biggest losers as they were crushed 8-1 at home by Mansfield, who trailed 1-0 for most of the first-half before running riot.
Among the others non-league sides to make it through were Stourbridge who beat Biggleswade Town 4-1 and Salisbury who eased past Dartford 4-2.
Grimsby and Scunthorpe played out a goalless draw in the Lincolnshire derby, while Brentford thrashed fellow west Londoners Staines 5-0.
Sheffield United put aside their league problems to beat fellow League One side Colchester 3-2, with Chris Porter grabbing the winner nine minutes from time.
Preston put six past Edgar Davids' Barnet without reply and Rotherham defeated fellow Yorkshire side Bradford 3-0 at the New York Stadium.
The dream is over for Luton, who made it as far as the fifth round of the competition last season, as they went down 2-1 at fellow Skrill Premier side Welling.
But MK Dons, who also advanced to the fifth round in 2012/13, are still in the mix after they beat Halifax 4-1.