Jangoo made 233 and Chase 194 in a partnership of 401 which eclipsed the record of 399 set by the England pair of Ben Stokes and Jonny Bairstow against South Africa 10 years ago.
Chase was dismissed just before the declaration on the third day of the first Test after reaching his sixth test century, his first in seven years and his highest career score.
Left to face four overs before stumps on Saturday (Sunday AEST), Sri Lanka lost Pathum Nissanka in the second over and were 1-15 at the close, still 303 behind. Kasun Rajitha was 4 not out and Nishan Madushka 2.
Nissanka has had a disappointing Test with scores of 2 and 3 coming off his 158 in his previous Test against Bangladesh.
The West Indies resumed on Saturday at 5-271, 37 runs behind Sri Lanka's first innings of 308. Jangoo was 78 overnight and Chase 42.
After coming together at tea on the second day, Jangoo and Chase batted through all of the first and second sessions on the third day before Jangoo fell in the middle of the final session.
Playing in only his second Test and as a stopgap for the injured Shai Hope, the Trinidad-born left-hander made light of his previous highest Test score of 30.
He reached his double century from 342 balls and only 116 balls after he posted his maiden century. That showed the trajectory of his innings; cautious at first but developing increasing confidence and fluidity. He hit 17 fours and a six.
After adding 85 runs from 25 overs in the first session on Saturday, Jangoo and Chase put on 136 from 27 overs in the second.
Chase was 175 when Jangoo was out, caught off a top edge from the bowling of Milan Rathnayake, one ball after hitting the same bowler for a towering six over deep backward square.
Chase carried on after his partner's dismissal and seemed set to match Jangoo's double century but was bowled by Sonal Dinusha six runs short of that milestone. Chase attempted to sweep a ball from the hard-working spinner but flicked it onto his leg stump.
The Sri Lanka bowlers had to toil on a flat pitch on Saturday in the absence of their attack spearhead Lahiru Kumara, who suffered a hamstring strain bowling his second over on the second day.
Milan Rathnayake played a major role in just his sixth test match, taking 5-124 from 35.5 overs — his first five-wicket bag. Dinusha was a workhorse, bowling 56 overs and taking 2-234.
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