The striker struck late in the second half to settle a hard-fought match.
Grant Leadbitter then got free down the left and clipped a cross to Jutkiewicz, but Charlton did just enough to clear the danger.
Charlton struggled to exert any real pressure on Boro and it was Tony Mowbray's men who continued to ask all the questions.
First Jutkiewicz turned in the box but was denied by Dorian Dervite and then Leadbitter tried his luck from long range but failed to test Hamer.
There was little in the way of chances for either side toward the end of the first half.
Charlton had the better of the early exchanges after the break but struggled to find a way past the Boro defence.
Jutkiewicz then had a clear sight on goal but shot straight into the arms of Hamer.
However, he rectified that a few minutes later to open the scoring. The Charlton defence failed to clear from a corner and after Hamer had come to collect the second ball, Jutkiewicz nodded home.
Chris Powell's side responded well with successive attacks, but both Green and Stephens missed good opportunities.
Yann Kermogant went close with a deflected effort but Boro held firm to secure the three points.