Colls have made it through to this stage for the first time in the club's 63-year history and are now just two more wins away from the first round proper when the Football League teams enter the competition.
They must first overcome Northern Counties East League side Handsworth FC, with the tie due to take place over the weekend of October 2.
It is the first time Ponte have been drawn at home in the competition this year and it will be their first home FA Cup tie for three years.
They earned their place in the third qualifying round with a come from behind victory in Blackpool after their hosts, Squires Gate, struck first with Dean Ing netting from close range.
More chances came and went for the home team before Colls finally got a foothold in the game and came close themselves with striker Eli Hey unlucky to hit the post.
Ponte were level on 18 minutes when Vaughan Redford broke from midfield and combined with Gavin Rothery, whose cross was headed in by Scott Smith.
They then went ahead just before half-time as Jack Greenhough got forward to cross for Hey to score from close range.
In the second period chances were more limited, but Redford headed one opportunity wide after meeting Jack Lazenby's cross.
Colls did extend their advantage on 82 minutes as Derry Robson charged forward from midfield and saw his shot go in via a deflection.
Squires Gate did not give up and caused a little late alarm when awarded a 90th minute penalty that was put away by James Boyd. But Ponte held on for a memorable victory that sparked some big celebrations among the management, players and supporters.