Early goals from Joel Appiah and Daniel Izekor put them on top at the break at Newham Leisure Centre.
But debutant goalkeeper Matt Knight kept Frenford in the hunt and an own goal halved the deficit to give them hope.
The visitors ended up empty handed, though, after Sam Cross missed a penalty.
Newham head to Hullbridge Sports on Bank Holiday Monday, while Frenford face an FA Vase tie at Benfleet on Sunday.
Meanwhile, Newbury Forest came out on top by the odd goal in seven after a 4-3 win at Thurlow Nunn League Division One South rivals Cannons Wood.
Forest head to Harlow Town in the FA Vase on Saturday.
Clapton claimed their first point of the season after a 2-2 draw at Park View, where Akiljo Balliu and Al Moataz Leguemani were on target either side of the break.
Tons visit Sheerwater in the Vase on Saturday, while Hackney Wick head to Ilford after suffering a last-minute 3-2 defeat against Dunmow Town, for whom Asa Cansdale netted a hat-trick in a fourth successive win.
Barkingside let a two-goal lead slip against Enfield Borough at Cricklefields and had to settle for a point from a 2-2 draw.
Michael James-Akerele netted a quickfire brace to put them on top, but Marco Magunga hit back with both goals for the visitors.
It enabled Barkingside to end a run of five successive defeats ahead of their return to action in a derby at Newbury Forest next Tuesday.
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