Raza and Moor came together with Zimbabwe reeling on 46 for four in their second innings, still needing 76 runs to make the West Indies bat again.
By the close of the fourth day they had carried the home side to 140 for four to claim an 18-run lead, with Raza adding his second half-century in a match where he has also claimed a five-wicket haul.
"We've got to get through a session and get ourselves into a position where we can maybe consider giving them a target," Zimbabwe coach Heath Streak said.
"But it's very tough, especially with the deficit we've had to make up, and with the pitch being so slow."
Zimbabwe's chances of ending a 10-match losing streak in Tests were badly dented by a 212-run partnership between Shane Dowrich and Jason Holder -- a record for the eighth wicket for the West Indies.
Dowrich's 103 was his first century in Test cricket, while Holder's 110 was his second.
"Shane has not had the best time in the recent past and was in need of a score, so it was good to be out there with him," said Holder.
"We were under pressure with the Zimbabweans bowling well. We really needed a partnership to get back in the game and it was down to me and Shane to put one up."
"They batted really well and put us under pressure, and we didn't help ourselves with a few dropped catches," said Streak.
"We've got ourselves to blame for giving them that lead, but we've just got to deal with where we're at now."
After spending 178 overs in the field, Zimbabwe's punch- drunk top order were easily knocked over as Kemar Roach grabbed two wickets in his opening spell.
But Raza made an unbeaten 58 and Moor batted 152 deliveries for his 39 not out to ensure that the series, which the West Indies lead 1-0, will go into a final day.
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