Chase (47 not out) and wicketkeeper-batsman Dowrich (27 not out) came together in the first over after lunch when opener Kieran Powell fell for 33 to a second slip catch by Younis Khan, reducing the home side to 71 for five and giving Mohammad Amir his third wicket of the innings.
On the same ground where an unbeaten hundred in just his second Test match saved the West Indies from defeat against India a year earlier, Chase again batted confidently against all bowlers.
Dowrich hardly looked troubled on a pitch that proved helpful to Pakistan's seamers in the morning session but appeared to lose much of its life in the afternoon.
Earlier, Mohammad Abbas had a dream start to his Test career after his captain, Misbah-ul-Haq, chose to bat first on winning the toss.
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He struck with his second delivery, having Kraigg Brathwaite caught at second slip, also by Khan, without scoring.
It represented important early success for the tourists as the opener had made history in their last Test match meeting with unbeaten scores of 142 and 60 that earned the Caribbean side a five-wicket win in Sharjah last November at the end of an otherwise miserable tour of the United Arab Emirates.
Maintaining a nagging accuracy, the left-arm fast bowler bowled both Hetmyer and Shai Hope, scattering the stumps with deliveries that jagged in sharply off a good length.
Pakistan's third seamer in their bowling armoury, Wahab Riaz, then removed Singh via a fine diving catch by Azhar Ali at backward square-leg to have the home side tottering at that point at 53 for four.
Notwithstanding his impressive entry to senior international cricket during the limited-over leg of the tour, the visitors resisted the temptation to include the uncapped Shadab Khan in their team as a second spinner in support of Yasir Shah, choosing instead to introduce27-year-old Abbas to Test cricket.
This is the 50th Test between the two nations and also the 50th Test match to be played at Sabina Park since the very first at the venue in 1930.