Burnley started the game the brighter of the two sides and enjoyed a sustained spell of possession without creating any clear openings.
But they fell behind when Dan Barlaser fed a neat ball through to Dijksteel who managed to lift his effort over the onrushing Trafford while falling over to give Boro the lead with his first goal of the season.
The Clarets responded well to going behind with Jeremy Sarmiento forcing Dieng into a good save at his near post and Jaidon Anthony firing a free-kick into the wall from the edge of the box.
Despite Burnley's pressure, the ever-lively Liverpool loanee Ben Doak came close to doubling the lead for Boro on the half-hour mark after wriggling through on the right-hand side and firing just wide of the far post.
The hosts eventually hit a deserved equaliser in the 37th minute when Roberts curled a left-footed effort over Dieng from 25 yards, with apparent assistance from the swirling wind brought by Storm Darragh.
It was Middlesbrough who came out looking more lively in the second period, however, creating several good chances within the first 12 minutes of the half.
An unmarked George Edmondson headed wide from the centre of the goal from six yards out before Tommy Conway and Delano Burgzorg both hit tame efforts straight at Trafford from inside the box.
Boro continued to dominate but could not make their pressure pay, with Emmanuel Latte Lath also excellently thwarted by Trafford as he tried to nudge it past the Burnley goalkeeper after racing through on goal.
Burnley improved in the closing stages as Anthony curled wide from distance, and Roberts failed to connect properly with a difficult close-range chance at the back post after a corner kick dropped to him at pace.
But despite their late rally, Scott Parker's team were made to settle for a point against a fellow promotion-chasing side, who will feel they were good value for the draw, having ended Burnley's run of four straight wins and five consecutive clean sheets.
The Clarets remain unbeaten at home but could drop out of the automatic promotion spots if third-placed Leeds United beat Derby County on Saturday.