Gareth Southgate's side dominated the first half and deservedly went in front five minutes before the break when Adam Johnson's corner found Sean St Ledger, who headed in from close-range.
In first-half injury-time Gary O'Neil's cross was only half-cleared into the path of Rhys Williams who buried the ball past the despairing Keiron Westwood.
Coventry were vastly improved after the break and deservedly got a lifeline with 12 minutes remaining when Best headed into the path of Clinton Morrison, who controlled well before firing across Brad Jones.
That set-up a tense finale and the home side's persistence paid off in the fourth minute of injury-time when Elliott Ward's cross found Best, who slotted home to send the Ricoh Arena into raptures.
Middlesbrough came flying out of the traps and Marvin Emnes raced through on goal but Keiron Westwood did well to make the stop.
However, Coventry came close to taking the lead when Morrison got away down the right and his cross found Michael McIndoe, who glanced a header just wide.
Emnes looked lively for the visitors and his 20-yard effort brought another good save out of Westwood.
The goal appeared to be coming for Boro and so it proved when Johnson's corner was met by St Ledger, who buried his header from six yards.
Coventry were rocked by going a goal behind and Emnes almost made it 2-0 before Williams made no mistake to put Boro in charge at the break.
Middlesbrough were once again the more lively side after half-time and only Westwood kept the game alive as he saved well from Johnson.
With Boro squandering their opportunities Coventry got into the game more and grew in confidence, with Best's mishit shot hacked away by David Wheater.
Minutes later, Sammy Clingan's cross found Stephen Wright in the box but he headed wide when he perhaps should have done better.
The Sky Blues were completely on top of the game and deservedly got back into the game when Best set up Morrison, who showed superb composure in the box before firing across Jones and into the back of the net.
Jordan Clark dragged an effort wide from the edge of the box and the game looked to have escaped Coventry until Best popped up with virtually the last kick of the game.
Ward found space down the left and drove in a low cross which found Best, who gleefully finished to earn his side a point.