Ollie Palmer opened the scoring in the first half against the run of play but Carl Dickinson cancelled that out with a goal three minutes before half-time.
Olufela Olomola did have the ball in the net on the stroke of the break as he headed in Jordan Green’s rebounded shot, but was adjudged to be offside.
Coming out much stronger in the second-half the home side had the better chances but it took something special from substitute Ashley Nathaniel-George for Crawley to retake the lead.
The defeat was confirmed in the last minute of stoppage time as captain Filipe Morias got his shot off from outside the area to inflict a first away defeat since the opening day of the season on Town.
Targeting a fourth successive away win, Darren Way made four changes to the side that drew 1-1 with MK Dons on Tuesday.
Rhys Browne and Jake Gray were in the starting XI for the first time in the league this season whilst Enes Mahmutovic was back in the side for his second game in Green and White.
Two of those changes were forced as Diallang Jaiyesimi and Omar Sowunmi came off in midweek to join an extensive injury list consisting of Stuart Nelson, James Bailey, Francois Zoko, Gary Warren, Wes McDonald and Alefe Santos.
Baked in early Autumn sunshine the visitors created the best chances in the opening exchanges with Green’s pace causing worry for the Crawley defence.
The winger hit the side netting for the first chance of the match and won a free-kick which was fired narrowly wide by Dickinson.
Making the Reds defence work early on, Alex Pattison played a ball over the top for Olufela Olomola which forced the attack wide, his shot from the angle was well saved by Glenn Morris.
Looking ever dangerous, Olomola drove forward with the ball and released Green, the Scunthorpe loanee looked for the return ball but Green opted to run it and the shot which eventually came from Browne was over.
The hosts managed to keep more of the ball as they grew into the first half and restricted chances for the Glovers, but they also failed to create any for themselves up until the 25th minute.
Determined work from Lewis Young down the right allowed a ball into the box. Luke Gambin seemed to have fluffed the chance but Palmer was there to stab the ball in at the back post to give Crawley the lead.
Matters got worst for visitors as Alex Pattison was forced off injured to become the next name on a seemingly ever-growing list.
However, despite the setback, the visitors found themselves level three minutes before half-time.
Olomola turned on the edge of the area and played in Dickinson who fired the ball past Morris to make it two in his last two on the road.
In first-half stoppage time, Olomola thought he had put the Green and Whites back ahead when he headed in virtually on the goal line from Green’s rebounded shot, but the flag was up from the linesman.
After the restart, the hosts started much better than they started the first, enjoying more of the ball in attacking areas.
Mark Randall had a shot from range which flew wide before Arquin had an effort saved at the other end following a swift counter-attack.
Looking threatening, Palmer should have restored Crawley’s lead just before the hour mark, finding himself free in the area he couldn’t manage to guide his effort on target.
However, Crawley did find themselves back in front thanks to a stunner from Nathaniel-George, with seemingly no danger on the midfielder unleashed a long range effort that flew past the helpless Baxter.
With a spring in their step, the home side looked to kill off the game and perhaps should have done through their two goalscorers.
When two-on-one against Enes Mahmutovic, Palmer opted to delay playing in Nathaniel-George till the last minute and his shot from wide of goal was stopped by Baxter.
Failing to create any real chances for an equaliser Yeovil could have found themselves further behind, Palmer had two bites of the cherry his first effort was saved by Baxter and the rebound was put wide with the goal at his mercy.
The defeat was confirmed in the closing minute, a good advantage from referee Neil Hair allowed Morias to get his shot away on the edge of the area which flew into the bottom corner, condemning Yeovil to a 3-1 defeat, their first away loss since the opening day of the season.
Crawley Town: Morris; Young, Vincelot, N’Gala, Maguire; Morias, Randall (Nathaniel- George 58), Payne, Gambin (Poleon 75); Camara, Palmer.
Bookings: Maguire 29’, Randall 32’, Palmer 85’
Subs not used: Mersin, Doherty, Francomb, Allarkakhia.
Yeovil Town: Baxter; James, Mahmutovic, Donnellan, Dickinson; Gray, Pattison (Arquin 34), D’Almeida, Browne (Fisher 63); Green (Patrick 79), Olomola.
Bookings: D’Almeida 32’
Subs not used: Phillips, Ojo, Mugabi, Rogers.