Theo Robinson saw his early penalty cancelled out by Lukas Jutkiewicz in the 19th minute, the Polish striker teeing up what would have been a seventh straight defeat for the Rams at the Riverside with an unchallenged header in the final stages.
But Boro failed to contain their opponents at the death, Coutts keeping Tony Mowbray's men tethered to mid-table until the weekend at least.
Grant Leadbitter warmed Frank Fielding's palms and Seb Hines cut out a dangerous pullback from Robinson as a frantic 45 minutes sparked into life.
Jutkiewicz should have made Boro's early pressure count, though, instead scooping Leadbitter's free-kick over the crossbar before Derby took the lead in the 16th minute.
A typically tricky Rams counter-attack saw Craig Bryson wade into the area only to tumble under the attentions of Andre Bikey, leaving Robinson to confidently tuck home the spot-kick.
But Boro were completely undeterred and levelled three minutes later. Richard Keogh conceded a free-kick well within Leadbitter's preferred range, the former Sunderland man seeing his low drive fumbled by Fielding, before raising a smile as Jutkiewicz bundled home the rebound.
The Polish striker nearly made it two after slicing Justin Hoyte's pullback wide but Derby had weathered the storm.
There were jeers from the away end as George Friend made a hash of converting Leadbitter's pinpoint delivery, but the travelling Rams were relieved when Luke Williams slammed the rebound across the goalmouth and out of play.
Rams boss Nigel Clough cut a frustrated figure as repeated raids were handled expertly by a well-drilled Boro defence - Jason Steele only getting a rare feel of the ball when Robinson squeezed a shot through the backline.
Soon, Bikey's long-range missile amused and enthralled in equal measure but it remained all-square at the break.
The sting had disappeared somewhat when play resumed, Jutkiewicz showing no bite when fed by Nicky Bailey before Coutts wasted a cross at the other end.
Bailey and Jutkiewicz soon jumped for the same delivery when one man would have sufficed and the ball bobbled away from danger.
A period of crisp passing followed - with no end product - but Robinson had the slippy turf to blame as he miscued over the bar on the hour mark.
Conor Sammon replaced the ineffective Nathan Tyson with Mowbray bringing on Merouane Zemmama and Emmanuel Ledesma for Kevin Thomson and Josh McEachran.
They soon proved handy, with Coutts denying Ledesma's rasping drive and Zemmama running out of luck following a mazy run as Boro sought a break in front of goal.
That burst of good fortune arrived 10 minutes from time through a familiar source, Jutkiewicz turning in time to notice Zemmama's forward chip and twisting his neck to guide the ball into the top corner.
Boro were up to eighth place as it stood, but in all the excitement, they let their guard slip at a critical moment, Coutts lashing home after a goalmouth scramble with just three minutes remaining.
And it would have been Derby's first victory at on Teesside since 2000 had Robinson tucked away the easiest of chances before the final whistle.