There is a 69-place gap between George Elkobi's Kent-based club and third-tier Stevenage.
But in their first third round appearance since 1988, the National League South minnows made history as Sam Corne netted a penalty in first half stoppage-time before fans flooded the pitch in celebration at the final whistle.
In Saturday's other early third round ties, Leicester took a break from their Championship promotion push to beat Millwall 3-2 at The Den.
Ten points clear at the top of the Championship, the 2021 FA Cup winners were able to rest several players and still saw off Millwall through goals from Cesare Casadei, Ricardo Pereira and Tom Cannon.
Ipswich, who are second to Leicester in the Championship, moved into the fourth round with a 3-1 win at League Two side AFC Wimbledon.
There were scares for Premier League sides Bournemouth and Brighton before they reached the last 32.
Bournemouth came from two goals down at half-time to win 3-2 at second-tier QPR via strikes from Marcus Tavernier, Kieffer Moore and Justin Kluivert.
Inspired by Joao Pedro's double, Brighton won 4-2 at Championship outfit Stoke despite falling behind early on.
Sheffield United, bottom of the Premier League, cruised to a 4-0 victory at fourth tier Gillingham.
Chesterfield, who sit top of the fifth tier National League, nearly stunned Championship club Watford, taking the lead at Vicarage Road before losing 2-1.
Ten-man Eastleigh, another of the four non-league teams in the third round, earned a creditable 1-1 draw at fourth tier Newport.
Later on Saturday, inconsistent Chelsea smashed Championship side Preston 4-0, while surprise title chasers Aston Villa edged second-tier Middlesbrough 1-0.
