Mowatt scored shortly after the hour mark before Dike made sure of the three points 15 minutes from time.
Barnsley sit fifth in the Sky Bet Championship and now hold a five-point advantage over Reading, who are just outside the play-off positions.
For Middlesbrough, it was a fourth consecutive away defeat.
The hosts started brightest in South Yorkshire, with Tykes front man Dike first to let loose on goal as he fired over Marcus Bettinelli's crossbar early on.
George Saville responded as he registered Boro's first chance of the afternoon. After failing to capitalise on Toby Sibbick's defensive error in the opening minutes, the midfielder caught the defender short at the back. Having intercepted a headed pass to goalkeeper Brad Collins, the Northern Irishman could not beat the goalkeeper from close range.
Barnsley's Cauley Woodrow then tried his luck with a volley that flew wide from inside Middlesbrough's penalty area midway through the first half.
Reds captain Mowatt was next to cause problems for the visitors, picking up the ball outside of the box and driving his effort high and wide.
Michal Helik forced Bettinelli to react quickly before the break. After claiming a deflected strike from the Barnsley centre-back, the shot stopper wasted no time in getting down low to meet the ball.
Paddy McNair then threatened the Barnsley defence after the restart but the only reward for his rocketed shot through a sea of red was a corner.
Mowatt broke the deadlock directly from a corner in the 62nd minute, the skipper's cross finding its way into the back of the net as Bettinelli rushed out and missed the ball.
An immediate second very nearly followed for Barnsley when Callum Styles' volley from a set play was kept out by Bettinelli.
Dike scored a fifth goal in five games to double Barnsley's lead in the 75th minute, netting a diving header after latching on to Callum Brittain's pass.
The forward was then eager double his personal tally as Barnsley applied the pressure, but a subsequent shot was blocked as the match drew to a close.