Scrum-half Haydn Thomas gave the visitors the perfect start with a try on five minutes and the Chiefs led throughout from there on.
The Scarlets, who saw Wales fly-half Rhys Priestland carried from the field with an ankle injury in the second half, did grab back a late penalty try from a scrum but it was not enough as they crashed to a third straight European defeat.
The Scarlets were without wing George North with a hip injury but six Wales squad members returned to their starting line-up including Jonathan Davies who captained the region.
Exeter made six changes with Argentina winger Gonzalo Camacho starting and centre Sireli Naqelevuki returning to the side.
The Scarlets went through the phases after Richard Kelly won the kick-off but it was the Chiefs who made their mark with their first-ever Heineken Cup try after five minutes.
From a scrum full-back Luke Arscott made a break through midfield and Thomas was on his shoulder to go over. Steenson added the extras.
But the Scarlets responded almost immediately with a 30 metre penalty goal from Priestland to make it 7-3 after eight minutes.
That Scarlets score was cancelled out on 12 minutes with a Steenson penalty after the home pack had stood up in a scrum on their line.
Both sides were forced into significant changes in personnel. Scarlets lost tight-head prop Lee and Chiefs replaced Thomas.
And Steenson was on target again to extend the Chiefs lead to 10 points after Scarlets were caught offside from an Aled Davies box kick.
The Scarlets got themselves in a good attacking position but knocked the ball on at a ruck with Andy Fenby isolated.
But from the resulting scrum the Scarlets earned a penalty on the 30-minute mark which Priestland converted.
Chiefs responded with a third Steenson penalty after the Scarlets found themselves under pressure at the scrum, while Priestland missed a penalty from inside his own penalty two minutes before the interval.
The Chiefs produced an impressive attack from the kick off but a knock-on five metres out prevented them extending their 16-6 interval lead.
The Scarlets suffered a real setback when Priestland was carried off after picking up an ankle injury when he slipped when in possession. He was replaced by Aled Thomas.
And it got worse for the home side as Steenson was on target with his fourth penalty goal on 49 minutes to restore a 13-point lead.
Aled Thomas kicked a fine long range penalty but that was cancelled out four minutes later with Steenson's fifth penalty shot as Chiefs led 22-9 on the hour mark.
But despite their lead the Chiefs were forced to defend their 22 and in the process lost prop Ben Moon to the sin bin for collapsing a scrum
And with the extra man the Scarlets earned a penalty try from French referee Mathieu Raynal on 67 minutes to cut the Chiefs lead to 22-16
The Scarlets went in search of the winning score but despite hammering at Exeter's 22 the visitors held on for a famous win.
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