A disappointing attendance of 15,033 at the Riverside Stadium - just 400 more than the record low Boro recorded in their last home match with Sheffield United - were becoming restless when Kink hit a 79th minute equaliser with his first touch.
The Estonia midfielder then produced a dramatic twist when he netted a free-kick deep into stoppage time.
Burnley had taken a 65th minute lead when Andre Bikey scored to cap a lacklustre spell after half-time from Strachan's men.
Chris Eagles and Jack Cork made their first Championship starts of the season for the Clarets after playing a telling role from the bench in Saturday's stunning 4-3 comeback win over Preston.
Strachan dropped Kink, Nicky Bailey and Kris Boyd from the Middlesbrough line-up that was humbled 3-0 at QPR.
Julio Arca, Andrew Halliday and Leroy Lita came into the starting XI, as did Justin Hoyte in place of the injured David Wheater.
And the much-changed side had a golden opportunity to open the scoring in the fourth minute.
Lita whipped in a dangerous cross that his strike partner Scott McDonald inexplicably scooped over from close range.
Middlesbrough had made a bright start to the match and McDonald drew a sharp save from Brian Jensen before blazing over once more in the 25th minute.
Burnley then created their first chance of note when weekend hat-trick hero Chris Iwelumo headed Ross Wallace's measured delivery beyond the crossbar.
Another testing centre from Wallace led to Clarets centre-back Bikey thumping a header against the post five minutes before half-time.
Halliday found space to meet Barry Robson's 51st minute corner but could not find the target with his header.
Moments later Jensen produced an eye-catching double save from O'Neill, the first a particularly impressive stop from a close-range volley.
But Burnley started to enjoy their best spell of the contest, inspired in large part by Eagles' inventive running.
Boro were penned in their own half and after 65 minutes the seemingly inevitable happened.
Graham Alexander delivered the Clarets' latest dangerous set-piece - they scored twice from corners against Preston - and this time Bikey made no mistake, thumping a header past Jason Steele.
As sections of the sparse crowd began to voice their discontent, former Celtic boss Strachan responded by introducing Kink for O'Neill with 15 minutes remaining.
It proved to be a masterstroke when he opened his Middlesbrough account shortly afterwards, drilling home a first-time shot from 20 yards.
In the third minute of injury time Kink became the toast of Teesside when he rifled in a spectacular 25-yard free-kick after Alexander had fouled Lita.