Saji first took the exam in January and scored 99.99 but, because he was not satisfied, he sat for the test again in April
They turned the game on its head with a counter-attacking partnership as Dowrich ended Tuesday unbeaten on a career-best 75, while Holder contributed 71 not out.
That saw the West Indies to stumps on 374 for seven, leading by 48 runs.
"Sometimes you're going to have to raise your hand and say that we tried everything and it didn't work because the opposition have batted well," Raza said. "Credit to them for batting that well."
The record partnership for the eighth wicket at Queens Sports Club took some of the shine off the offspinner's five for 82, which was the first five-wicket haul by a Zimbabwean in Tests at the ground since 2003.
Having contributed 80 in the first innings, Raza was also the fifth Zimbabwean to score a 50 and take five wickets in a Test.
While the second evening had produced attritional cricket as the West Indians crawled to stumps on 78 for one, the tourists were more adventurous on day three with opener Kieran Powell leading the way in a 55-run stand for the second wicket with nightwatchman Devendra Bishoo.
It took the introduction of Raza to change the course of the morning, as he had Bishoo caught and bowled for 23 and then trapped Kyle Hope lbw for one.
Powell looked destined for a fourth Test century after he was dropped on 85 by Graeme Cremer, but seamer Chris Mpofu ensured the mistake was not costly as he used the second new ball to have the opener caught at gully for 90.
An attacking 56-run stand for the fifth wicket between Shai Hope and Roston Chase put the West Indies back on top after lunch, before Raza intervened once again.
After Chase was trapped lbw for 32 by the offspinner, Hope was bowled for 40 and Jermaine Blackwood became Raza's fifth victim.
With a large deficit and just three wickets in hand the West Indies looked set to concede a first-innings lead on a difficult pitch, but Dowrich and Holder swung the match back in their favour.
"We're only 48 runs ahead, but we've put lots of miles on their legs," said Powell.
"We're going to come back tomorrow with two set guys, put some more miles on their legs and stretch out that lead as much as possible. Any lead is a good lead on this pitch."
West Indies first innings
K. Brathwaite c Masakadza b Cremer 32
K. Powell c Ervine b Mpofu 90
D. Bishoo c and b Sikandar Raza 23
K. Hope lbw b Sikandar Raza 1
S. Hope b Sikandar Raza 40
R. Chase lbw b Sikandar Raza 32
J. Blackwood c Cremer b Sikandar Raza 5
S. Dowrich not out 75
J. Holder not out 71
Extras (2b, 2lb, 1nb) 5
Total (7 wkts, 150 overs) 374
Fall of wickets: 1-76 (Brathwaite), 2-131 (Bishoo), 3-135 (K. Hope), 4-163 (Powell), 5-219 (Chase), 6-225 (Blackwood), 7-230 (S. Hope).
Still to bat: K. Roach, S. Gabriel.
Bowling: Mpofu 21-7-46-1, Mire 5-2-5-0, Cremer 43-7-134-1, Sikandar Raza 43-12-82-5, Chisoro 34-7-94-0, Masakadza 4-1-9-0.
Toss: Zimbabwe
Umpires: Kumar Dharmasena (SRI) and Simon Fry (AUS)
TV umpire: Paul Reiffel (AUS)
Match referee: Javagal Srinath (SRI)
Series: West Indies lead two-match series 1-0.
Saji first took the exam in January and scored 99.99 but, because he was not satisfied, he sat for the test again in April
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