The Moose advance to face either Grand Rapids or Hamilton in the second round.
Down 2-1 heading into the third period, Manitoba got a goal from rookie Matt Pope at the 1:45 mark before Sawada banged home the rebound of a Pope shot to give the Moose the lead.
Guillaume Desbiens iced the win with his third goal of the series at 15:46.
The Moose outshot the Marlies 11-5 in each period and 33-15 for the game en route to their third win in a row.
Tim Stapleton and Alex Foster scored for the Marlies and Adam Munro, making his first appearance of the postseason after backing up Justin Pogge in the first five games, stopped 29 of 33 shots in defeat.
Attendance at the MTS Centre was 10,947 — the largest crowd for a first-round AHL playoff game since 1998.