Ramharack's spell restricted Bangladesh to 8-103, before West Indies skipper Matthews led the chase with her 22-ball knock that featured six boundaries, to help her side reach their target on 2-104 in just 12.5 overs.
With their second win in three games, the West Indies have boosted their hopes of reaching the semi-finals, moving atop group B, but they may well need another win in their final group game against England to qualify for the semis.
England have a game in hand, as they play Scotland on Sunday, and then West Indies on Tuesday.
Second-placed South Africa also need a win in their last group game against winless Bangladesh on Saturday to advance.
Ramharack was in action early, getting Bangladesh opener Shathi Rani stumped for nine and Dilara Akter bowled on 19.
Shobana Mostary (16) revived the innings in a 40-run partnership with skipper Nigar Sultana, who top-scored with 39 off 44 balls.
But Ramharack again provided the breakthrough, with Mostary stumped in the 13th over.
Bangladesh slumped as leg-spinner Afy Fletcher took 2-25.
Sultana hit four fours but couldn't generate any more momentum with the lower middle order, and Bangladesh had to settle for a below-par total despite a slow Sharjah track.
In reply, Matthews brought her power game as she and Stafanie Taylor smacked 52 off the first 45 balls.
Matthews provided the punchy start before she was bowled in the eighth over.
Taylor scored 27 off 29 balls, including three fours, before retiring hurt.
That didn't stop the West Indies, though, as Shemaine Campbelle scored 21 off 16 balls and Deandra Dottin an unbeaten 19 off seven balls, including two sixes.
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