The Iron started the game without their first-choice centre-backs, with David Mirfin and Paul Reid both absent through injury.
Reid sat out as a precaution following the broken bone in his hand, though he may well have featured if it was a competitive game. Mirfin's absence came as a surprise.
Trialist Tom Newey, as good as certain to be offered a deal, partnered Niall Canavan at the back in their absence.
The visitors took the lead with the first real chance of the game after seven minutes. Cameron Park cut in from the left and fired past Sam Slocombe from 20 yards.
That was a blow to the confidence of the Iron, and the visitors argued in vain for a penalty nine minutes later when a shot came of Christian Ribeiro. They had to settle for a corner.
The Iron responded, with a Mike Grella shot being turned away by Boro keeper Jason Steele using his legs. It was the home side's first real chance of the game.
But that did not prompt an Iron fightback, and Boro thought they had doubled their advantage when Scott McDonald pounced from close range. The former Celtic forward was offside, so the score remained the same.
That single-goal advantage was soon two, however, when Canavan felled Curtis Main in the area.
Main converted the spot-kick by sending Slocombe the wrong way, and it looked the the Teesiders had a cricket score on their minds - with Andy Halliday smashing over and Main firing wide.
The Iron gave themselves a lifeline right on the stroke of half-time when Grella chipped Steele, ensuring that for all the dominance of the visitors they only held a one-goal advantage at the break
Connor Ripley came on to replace Steele in goal for Boro, and he was beaten 12 minutes into the second-half.
Josh Walker cut inside from the right and hit the ball past Ripley into the top left corner.
Chopping and changing from both managers disrupted the flow of the match, but Grant sealed victory when he smashed the ball into the top left corner from 25 yards out.
Matt Godden hit the crossbar with a further effort in stoppage-time, from Connor Jennings' low cross, but victory was already assured.