The 20-point victory puts them 10 points from safety with three fixtures remaining.
Bracknell skittled out the visitors for just 79 and Hafiz Mahmood (59*) led the way with the bat once again to see his side home without the loss of any wickets.
Finchampstead had three players reach half centuries as they recorded a 41-run victory over Henley 2nds in Thames Valley Division One.
Matt Jones (82), Russell Lane (73) and Andy House (51) all contributed to an imposing target of 284-4 set by Finch.
Sam Taylor was the chief wicket taker, finishing with figures of 6-73. Henley battled hard with Andy White (83) and Jason Barber (50) both reaching half centuries before their side were bowled out for 243.
In the same division, Wokingham took less than half their allotted overs to reach their target in a seven-wicket win over Amersham.
The victory moves Wokingham up to joint-second with High Wycombe 2nds but still some way behind leaders Reading, who only drew with Kew.
The hosts bowled out their opponents for 113 and then ran down that total in 21.5 overs with seven wickets to spare.
Eversley kept up the pressure on Division 2a leaders Datchet with a 20-point victory over North Maidenhead.
Chris Last was the stand-out bowler for Eversley, taking six wickets for 58 runs to help dismiss the hosts for 210.
Scott Dyer put on an unbeaten knock of 77 and was well supported at the crease, as Eversley reached the target with eight wickets in hand.
The win puts them just 20 points behind Datchet ahead of the crucial clash between the two this weekend.
Royal Ascot's poor form continued after a heavy 125-run defeat to Chiswick & Latymer in Division 3a.
Chiswick made 206 from their allotted overs but Ascot were skittled out for 81.