Fly-half Freddie Burns grabbed an early try and kicked nine more points, while Akapusi Qera and Olly Morgan also crossed the try-line for the visitors.
It was a disappointing performance from Exeter, who were made to pay for a lack of composure despite scoring tries through Hoani Tui and Matt Jess.
The hosts took the lead in the second minute as Gareth Steenson slotted over a 35-metre penalty, but Gloucester hit back quickly as Burns fended off a couple of defenders before touching down.
The fly-half added the simple conversion, before the Chiefs squandered a chance to respond when Fijian centre Sireli Naqelevuki was held-up over the line.
Exeter were in the ascendancy, however, with left wing Jess coming within five metres of the line before Tui powered over for the score after several further phases of play.
They were ongoing problems in the scrum and Steenson slotted a 40-metre kick between the uprights to give the Chiefs a four point advantage after Gloucester prop Rupert Harden was penalised.
But Exeter then gave away a soft try as Mike Tindall and Eliota Fuimaono-Sapolu linked-up with flanker Qera, who stepped out of a Gonzalo Camacho tackle to cross the line for Gloucester's second try, which was converted by Burns.
That gave Gloucester a 14-11 lead at half-time and they continued to dominate after the break, although they initially failed to capitalise as Burns missed a penalty.
However, the visitors kept the pressure on the Exeter line and full-back Morgan squeezed in the left corner with Burns adding the extras from out wide.
At the other end, Exeter were wasting their opportunities when they got inside the Gloucester 22, turning down two kickable penalties as the visitors mounted a stout defence.
Problems continued in the front rows in and Exeter were awarded a penalty which Argentina international Ignacio Mieres, who had replaced Steenson, made no mistake with.
With 12 minutes remaining, Exeter carelessly threw the ball around in their own 22 and Gloucester were awarded a penalty which Burns duly scored to make it 24-14.
Gloucester went in search of a bonus point but it was Chiefs replacement scrum-half Kevin Barrett who broke through before playing in Jess to hand Exeter an extra point.
However, that failed to take the gloss off the victory for Gloucester, who were left to celebrate their first away win since they beat Northampton 18-16 last season.
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