Perhaps a reflection of the importance of the meeting for both sides, the opening 45 minutes provided little in front of goal.
Ryan Seager forced the only meaningful save of the half with his shot from close range as the deadlock was rarely threatened.
Teeing up a second half with it all to play for Adel Gafaiti headed just wide on 46 minutes before the visitors opened the scoring as the final quarter-of-an-hour approached.
Filipe Morais was perfectly placed at the back post to sweep home a neat volley which proved to be the difference, as Town’s survival hopes were dented.
Buoyed by last weekend’s dramatic draw at Swindon, Neale Marmon made two changes to the side as he switched to 4-4-2.
The last-minute hero in Wiltshire, Gafaiti took the place of the injured Omar Sowunmi in the heart of defence whilst Seager was given his first start since rejoining the club in January.
Amongst an attacking looking bench, Alex Pattison returned after a three-game absence with a niggling back problem to provide a positive before a ball had been kicked.
Once underway, and with the Middlesbrough loanee awaiting his chance on the sidelines, a match of vital importance for both sides began in uneventful manner.
Perhaps keen to avoid conceding early on, chances remained at a premium as the opening quarter-of-an-hour passed with neither ‘keeper called into action.
Tristan Abrahams’ wayward strike from distance and Reece Grego-Cox’s header into the side netting at the other end provided the only early entertainment before a bright spell from the hosts.
Ticking past the 20-minute mark, Bevis Mugabi produced the game’s first shot on target with a curling shot into the arms of Glenn Morris, swiftly followed by a smart save from the Reds stopper.
Abrahams’ cross found Seager in the middle who quickly fired goalwards at the near post, but Morris was quick off his line to narrow down the angle and parry away the strike.
After a quick flurry of action, the cagey nature of the meeting in Somerset re-emerged to see the visitors enjoy more of the ball with little penetration.
Unable to test Nathan Baxter in the home goal the green and whites drew Morris into action again with the last act of the half as Tom James’ 25-yard free-kick had to be gathered at the second attempt.
Needing more urgency in attack, the Glovers almost took the lead within 60 seconds of the restart.
Winning a corner, Abrahams’ delivery was perfect for Gafaiti who rose highest in the middle but his glanced header fell inches wide to deny the Algerian a second goal in as many games.
A more open affair than the first-half, Crawley’s top scorer Ollie Palmer had the chance to test Baxter five minutes later but could only blaze over the bar from the edge of the box as quality in the final third remained lacking.
In the search for more of a cutting edge, top scorer Alex Fisher replaced Duffus just before the hour-mark but it was the visitors’ substitute Ashley Nathaniel-George who threatened goal as his low drive forced Baxter into his first save of the game with 25 minutes remaining.
A rare shot on goal for the Reds, Gabriele Cioffi’s side grew confidence from the effort and stunned Huish Park with the opener on 72 minutes.
Palmer’s ball to the back post perfectly picked out Morais who had last Daniel Ojo, with the winger coolly side-footing into the far corner.
Leaving little time to find a way back into the match, Town were left requiring late drama for a second successive match and it almost came on 85 minutes.
Pattison’s ball into the middle found the head of Seager, but from point-blank range, Morris was able to keep out the effort and dive on the ball with the striker fishing for a rebound.
Throwing everything at the visitors, the sense it wasn’t Yeovil’s day was confirmed when Morris kept out both Alex Fisher and Sessi D’Almeida in the dying embers of the match to condemn the green and whites to defeat.
Yeovil Town: Baxter; Mugabi, Warren (Pattison 68′), Gafaiti, Ojo; Abrahams, Worthington, James, Dobre (D’Almeida 83′); Duffus (Fisher 57′), Seager.
Bookings: Warren 39′, D’Almeida 90′
Subs not used: Nelson, Sowunmi, John, Rogers.
Crawley: Morris; Young, McNerney, Dallison, Maguire; Bulman; Grego-Cox (Nathaniel-George 54′), Camara (Willock 68′), Morais (Gambin 86′); Francomb; Palmer.
Bookings: Morais 60′, Nathaniel-George 63′, Young 78′
Subs not used: Mersin, N’Gala, Sesay.