Resuming on 298-8, Gareth Berg finished on 72 and Ryan McLaren made 52 as Hampshire were bowled out for 326,
Despite Brendan Taylor's 90, Notts went from 127-2 to 185 all out, Edwards (4-57) ending on match figures of 10-145.
Needing 200 to win, Hampshire closed on 89-0 with Michael Carberry unbeaten on 37 and Jimmy Adams 46 not out.
There is plenty on the line at Trent Bridge going into the final day. Hampshire could secure safety with a victory but only if Yorkshire beat Sussex, who would be relegated instead.
Meanwhile, with Middlesex losing inside three days to Worcestershire, Notts know victory for them would secure second place in Division One, although Hampshire remain firm favourites after dominating day three.
Hampshire began day three on 298-8 and Harry Gurney (4-70) eventually broke Berg and McLaren's 123-run ninth-wicket partnership, getting the latter to play-on to a fuller ball.
An over later, Stuart Broad (2-73) had Berg caught down the leg-side by Chris Read - his fifth catch of the innings.
Taylor was the mainstay of the Nottinghamshire's second innings and passed 1,000 first-class runs in the process.
But Edwards tore the heart out of the Notts middle-order and apart from Taylor, only Steven Mullaney (19), Samit Patel (25) and Billy Root (15) reached double figures.
McLaren took 2-22, while Berg (2-28) also impressed with a couple of great deliveries to get rid of Samit Patel and first-innings centurion Riki Wessels.
Carberry and Adams then then went after the Notts bowling, with England bowler Broad going at more than eight an over.
The opening pair took Hampshire safely to the close, needing another 111 runs to win to stand a chance of Division One survival.
Hampshire fast bowler Fidel Edwards' 10-wicket match haul against Nottinghamshire at Trent Bridge boosted his side's hopes of survival.