Jenkins has come under pressure following his team's disappointing start despite his recruitment drive during the summer. But it was local products Aled Gravelle and Lee Byrne as well as summer signing Inoke Afeaki who scored the tries to relieve the pressure.
Hooker Gravelle pounced on the stroke of half-time after Connacht outside-half Paul Warwick rattled the Scarlets' nerves with a stunning solo try on 18 minutes.
Gareth Bowen then gave the home side the lead 90 seconds into the second half with a penalty before Afeaki started and finished their second try. The Tongan giant won a lineout 25 metres from Connacht's line but the Irish province could do little to stop a devastating drive from the Scarlets pack.
Byrne then put the Scarlets clear with a class finish from his own hoisted kick. The rookie full-back out-jumped the Irish defence and cantered over six minutes from time. Replacement Chris Venter crashed over late on for Connacht but the Scarlets had done enough for their second win of the Celtic League season.
Tries - A Gravelle, I Afeaki, LByrne
Cons - G Bowen (2)
Pens - G Bowen (2)
Tries - P Warwick, C Venter
Cons - P Warwick, D Slemen
Drop G - D Slemen
Referee: D Changleng (Scotland)