Spalding pipped lower level Tividale 1-0 thanks to a second-half goal from in-form Stefan Broccoli following a cross from substitute Marcus Maddison.
Stamford eased to a 2-0 win at home to higher level Redditch United with Cosmos Matwasa and Ed Cook.
Spalding and Stamford meet in a mouthwatering Northern Premier League derby at the Sir Halley Stewart Field on Tuesday.
Wisbech will try and book their place in the second qualifying round draw when they travel to Baldock on Sunday.
In the United Counties Premier Division North a second-half Will Bird header delivered a 1-0 win for Deeping Rangers against Melton Town at Outgang Road, while injury-hot Holbeach drew 0-0 at home to Pinchbeck United.
Peterborogh Northern Star were beaten 5-0 at home by Bugbrooke St Michael in the Premier Division South, but there was a
first Thurlow Nunn Premier Division win of the season for March Town who beat Whitton 2-0. Jack Friend and Craig Gillies scored the goals. The Hares are out of the relegation zone after picking up their first points of the campaign.
Whittlesey Athletic are up to fourth in Division One North after a 3-2 success at Needham Market Reserves. Peterborough North End are in action at home to Wisbech St Mary on Tuesday (September 7).
FA Cup first qualifying round: Peterborough Sports 2 (Stevenson, Jarvis), Haringey Borough 1; Tividale 0, Spalding United 1 (Broccoli); Stamford AFC 2 (Matwasa, Cook) Redditch 0.
Sunday fixture: Baldock Town v Wisbech Town,
United Counties League
Premier Division North: Deeping Rangers 1 (Bird), Melton 0; Holbeach 0, Pinchbeck 0.
Premier Division South: Peterborough Northern Star 0, Bugbrooke St Michael 5.
Division One: Blackstones 0, Dunkirk 2; Hucknall 3, Bourne 0.
Thurlow Nunn League
Premier Division: March 2 (Friend, Gillies), Whitton 0.
Division One North: Parson Drove 0, Huntingdon 3; Whittlesey 3, Needham Market Res 2.