Goals from Jonny Howson and Chuba Akpom either side of half-time gave the Teessiders all three points before cauley woodrow scored a late consolation goal from the penalty spot.
Our Boro writer Joe Nicholson has dished out his marks following the match – scroll down and flick through the pages of our picture gallery to see them:
9. George Saville - 7
Snapped into challenges and made some important blocks in midfield when Barnsley were playing some neat football. 7 Photo: Alex Davidson
10. Chuba Akpom - 7
Missed a good chance early on but was always alert and looking to run in behind. Reacted quickly to score his second goal in as many games. 7 Photo: MFC
11. Britt Assombalonga - 6
Another hard-working display even if the ball didn’t quite drop for him in the final third. 6 Photo: Getty Images
12. Hayden Coulson - N/A
(for Akpom, 86) Came on in the closing stages. N/A Photo: George Wood