James Beattie's side then weathered some heavy second-half pressure before snatching victory through Piero Mingoia with nine minutes remaining.
Boro started brightly and went close after four minutes as Grant Leadbitter squared for Marvin Emnes but he could only fire against the legs of the goalkeeper. Then from the resulting corner Jutkiewicz sent a glancing header wide.
Boro were quickly back on the attack to grab the opener after nine minutes as Jozsef Varga headed back across goal for Jutkiewicz to deftly chip past the advancing keeper and then slot into an open goal from six yards.
Stanley levelled after 38 minutes when Carver's looping header crossed the line.
Jutkiewicz, Mustapha Carayol and Emnes all went close in the second half but Boro were stunned in the 81st minute as Mingoia shot through a crowd after the hosts failed to clear a corner.
Boro launched a late assault and substitute Adam Reach stabbed just wide before Varga sent a low shot wide and in stoppage time and Leadbitter sent the keeper full length to fingertip around the post.