Jangoo reached a maiden half-century in only his second Test and was 78 not out at stumps.
Chase was unbeaten on 42 in a partnership worth 103 runs which carried the West Indies to 5-271 at stumps in reply to Sri Lanka's first innings of 308, a deficit of 37.
Until Jangoo and Chase came together in the final session, a Sri Lanka attack depleted by the loss of fast bowler Lahiru Kumara to a hamstring injury managed to keep chipping out West Indies batters, giving the visitors an early advantage.
Kumara clutched his hamstring while running in to deliver the first ball of his second over of the day. He hobbled off and didn't return before stumps.
In his absence, Asitha Fernando and Milan Rathnayake both compensated, taking two wickets each during a second session which belonged to Sri Lanka.
The hosts were in a strong position at 1-89 at lunch but went to tea at 5-168.
Openers John Campbell and Brandon King were responsible for the solid start, sharing a partnership of 58.
King's was the only wicket to fall in the first session, caught at short cover off Rathanayake when he was on 31.
Campbell was out early in the second session, holing out at long-on from the bowling of Sonal Dinusha when he made 37.
With the Sri Lanka bowlers applying pressure, runs dried up and wickets began to fall regularly. Rathanayake bowled Kavem Hodge off an inside edge for 16 and Fernando dismissed Joshua da Silva (20) and Justin Greaves (3).
Jangoo and da Silva put on 52 for the fourth wicket in a partnership which saw the West Indies begin to break the constraints of accurate bowling in the off stump channel.
Greaves tried to last until tea but fell to a superb, one-handed catch by wicketkeeper Kusal Mendis.
The left-handed Jangoo was cautious at first but increasingly used his feet and found ways to thread the ball through gaps on the off side.
With Chase, he kept a slow but steady scoring rate through the final session which they negotiated safely, strengthening the West Indies' hand.
By stumps, Jangoo faced 174 balls for his 78 and hit seven boundaries. Chase offered him stalwart support and took his 42 from 105 deliveries.
Fernando was both accurate and miserly, allowing only 25 runs from his 18 overs while capturing two wickets. Rathnayake was equally hard working and had 2-45 from 19 overs.
Both teams are playing their first matches after long breaks from Test cricket. Sri Lanka are playing only their third match in this World Test Championship cycle.
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