Boro moved up to eighth with the victory that started with Doak’s first goal for Boro.
The 18-year-old, making his first start since arriving on loan from Liverpool, pounced to convert from close range after goalkeeper Viktor Johansson had only palmed Riley McGree’s effort into his path.
That arrived 11 minutes before half-time and was followed by Hayden Hackney’s brilliant second with 17 minutes remaining.
The highly-rated midfielder’s run from deep at the Stoke defence was followed by him rounding former Middlesbrough defender Ben Gibson and firing into the top right corner.
Carrick said: “The allround team performance I thought was really good, we controlled a lot of the game and also looked dangerous.
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