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 goalkeeper Marcus Bettinelli rushed out and missed the ball.
Dike then 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 delivery.
Ismael said: "At the end of the day, all that matters is getting the job done.
"The first half was a sleepy game without intensity. The opponent was very organised and tried to take the time from the clock.
"We didn't find a way to put the pressure on the opponent in the first half, but we were stable and we didn't concede any shots.
"In the second half we changed to have more impact on the front three. We controlled the second half completely and, as usual, set-pieces made the difference.
"It's very important that we scored the second goal in the right moment. It was our purpose to win the game because the next three points mean that you cross the 70-point line.
"We are capable to score at any time and a big focus from the guys was to stay in the game, avoid mistakes and to apply pressure on the opponent.
"Key players make the difference at the moment and this is what you need in our situation.
"Dike is a player who has a big impact on the centre-back. It's unpleasant to play against him.
"It's another great performance and you feel the strength from the guys."
Middlesbrough boss Neil Warnock said: "If we took one of our chances then we would have won the game.
"The longer the game went on, you knew they were going to get something. What can you say about a goal like that (Mowatt's)?
"I've just apologised to Jonny Howson because I thought he was absolutely man of the match. He covered every blade of grass and he didn't deserve to be on the losing side today.
"We need to improve the squad and today demonstrated where we need to improve.
"I can't fault the effort. They battled to them and I don't think some teams fancy coming out against Barnsley.
"I was disappointed with a couple of the lads, but the two centre-halves were fabulous. Jonny Howson was out of this world and an absolute credit in a horrible game today.
"I don't think they had any idea how to break us down.
"Barnsley deserve everything that they get for the maximum amount of running that they do. I don't think some of my lads would enjoy playing in that team.
"There's no divine right and I hope they go up because they can ruffle a few feathers.
"I thought we should have won the game, but we ended up getting beat. You have to learn from these things and hope we can rectify it in the next couple of months."