Wellington Masakadza, another debutant, also shone with two wickets as Zimbabwe made quick work of Bangladesh's lower order, dismissing the hosts for 169.
Zimbabwe sensed a win once Raza claimed three vital wickets in the opening session to reduce Bangladesh's innings to half.
Kyle Jarvis and Mavuta joined the act with Raza in the morning as they chipped in with a wicket apiece to leave Bangladesh 111-5 at the break.
Bangladesh resumed play on 26-0 but within half-an-hour had lost their first wicket, with others following quickly.
Play started 30 minutes early to make up for the lost time after bad light brought a premature end to day three, with Bangladesh still needing 295 runs on a wearing pitch.
The umpire gave Liton not out but a TV replay indicated the ball would have hit the stump after pitching outside off -- ending Liton's innings on 23.
One down, Mominul Haque dragged a Jarvis delivery onto his stumps for nine before Raza bowled Imrul Kayes for 43 to leave Bangladesh at 83-3.
Raza then took a spectacular catch at cover on the stroke of lunch to hand debutant Mavuta his maiden Test wicket and send back Nazmul Hossain.
"I think the wicket was good, but we didn't apply ourselves with the bat. We were not disciplined enough to restrict our strokes," said Bangladesh skipper Mahmudullah. Mavuta dismissed Bangladesh's last accomplished batsman Mushfiqur Rahim soon after play resume before Zimbabwe wrapped things up.
The second Test will be held in Dhaka from November 11-15.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)