Asked to bat, opener Warner top-scored with a 41-ball 58 to lift Sunrisers to a modest 150 for six.
The bowlers then produced a disciplined effort to restrict Kings eleven to 130 for nine in 20 overs, paving the way for the visiting side's third win of the season.
Boult returned with impressive figures of 3 for 19 in his four-over spell while Bhuvneshwar Kumar chipped in with two wickets.
