Despite the sides having little to play for, the encounter was played at a high tempo throughout with plenty of attacking intent on display.
That endeavour reaped reward as early as the 10th minute when the Sky Blues took the lead through Lukas Jutkiewicz's eighth goal of the season.
But Boro have started to provide more threat under manager Tony Mowbray in recent times and it was evident soon after as Julio Arca levelled things up.
That seemed set to be the final goal of an enthralling encounter before Kink's late heroics earned Mowbray's men all three points.
Boro started positively, with Leroy Lita using his vast reserves of pace to create an opening before high and wide from 25 yards.
Coventry, though, also seemed determined to impose themselves in the early stages.
Indeed, their fluent play caused Mowbray's side problems in the opening stages and it came as no surprise when they took an early lead through Paul Smith.
Clive Platt was the orchestrator, cleverly working his way into the area before chipping over an on-rushing Paul Smith to leave Jutkiewicz with the easiest of finishes.
Middlesbrough hit back well and began to control proceedings with Andrew Taylor going close from 25 yards.
They got the reward their endeavour deserved in the 25th minute as Arca equalised with his second goal in as many games.
Lita deserved much of the credit, though, after jinking his way down the left wing and sliding a clever ball to Arca, who, taking advantage of some woeful defending, unleashed an excellent 20-yard strike past Keiren Westwood.
The game continued to be played at a fast pace, although both sides struggled to fashion any other opportunities before the interval.
The home side controlled proceedings after the break, though, with Marvin Emnes twice seeing goalbound efforts blocked by Jordan Clarke.
Westwood then had to be on high alert to deny Barry Robson, while Smith was on hand to produce a comfortable save from Jutkiewicz at the other end.
The end-to-end nature of the game was continuing as, first, Middlesbrough skipper Robson forced a fine save from Westwood before David Bell saw an effort sail just wide for Coventry.
Lita thought he had won it for Boro late on, but his effort was ruled offside.
And, just when the game seemed to be petering out into a draw, the home side claimed all three points as Kink curled an excellent left-foot effort past Westwood.