Chasing a target of 153, Banton was the glue that held the innings together as England lost early wickets as Scotland felt an upset was on the cards.
England held their nerve though as Banton found good support down the order from Sam Curran (28) and Will Jacks (16 not out) to get over the line with 10 balls to spare.
It was a bit different at the start when opening pair Jos Buttler and Phil Salt were dismissed within the first two overs to be reduced to 13-2.
Jacob Bethell played nicely for 32 but the left-hander's dismissal triggered another mini-collapse as captain Harry Brook also fell cheaply.
Banton's runs came off just 41 deliveries as England will be relieved with the win following a defeat against West Indies in their previous encounter.
Earlier, Scotland lost wickets at regular intervals to be 153 all out as England spinner Adil Rashid took 3-36 in four overs.
Captain Richie Berrington top-scored for the Scots with 49 from 32 balls but there was not enough bigger contributors in the batting order.
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