The third-placed Swans had been guilty of wasting a series of chances but Sinclair, called into the England Under-21 squad earlier this week, cut in from the left flank in the 85th minute to fire his 12th goal of the season past Jason Steele with the help of a deflection.
The goal was just reward for the Welsh outfit, who were always in control and have gained ground on second-placed QPR in the Championship table.
Middlesbrough barely registered as an attacking threat and remain just above the relegation places on goal difference alone.
Swansea brought Joe Allen and Mark Gower back into their midfield following the disappointing defeat against Bristol City in midweek, which ended their unbeaten home record, and Craig Beattie was given a chance in attack with Marvin Emnes unavailable to face his parent club.
Middlesbrough left Kris Boyd on the bench after his goalscoring exploits against Scunthorpe, with Scott MacDonald and Leroy Lita getting the nod up front. Justin Hoyte started on the right.
Swansea made a bright start and they came close to a fourth-minute opener as Darren Pratley overlapped down the right and fired the ball across the six-yard box, but Sinclair could not make firm enough contact with the goal at his mercy.
Allen fired wide with a speculative effort with the hosts almost monopolising possession, and it needed a good block from David Wheater to deny another Allen strike.
Ashley Williams headed over from a Gower corner and Boro keeper Steele saved well from Dyer after the winger linked up with Pratley as the Welsh side continued to pile on the pressure.
Captain Gary O'Neil blazed over from Boro's first meaningful attack before the visitors difficult start was compounded when midfielder Kevin Thomson limped off after 20 minutes, Nicky Bailey his replacement.
Matthew Bates diverted a Beattie cross wide with Sinclair lurking although Boro did enjoy a couple of decent counter-attacks, with Lita looking sharp.
The best chance fell to Allen after Sinclair had skinned Bailey down the left, but the midfielder's shot was well saved by Steele and Angel Rangel's follow-up effort was blocked over the bar.
Middlesbrough's best chance came at the start of the second half with O'Neil's volley from a poor Williams header being blocked by Garry Monk.
Alan Tate and Barry Robson picked up bookings for ill-timed challenges on Bailey and Monk respectively as Boro slowly began to establish a foothold in the match.
On the hour mark Tate tricked his way into the Middlesbrough box and found Sinclair, but the flag was raised for offside.
Tony Mowbray sent on Julio Arca for Robson and Rodgers took off Gower and replaced him with David Cotterill with the game in desperate need of some inspiration.
The hosts were forced into a defensive reshuffle as skipper Monk departed with an injury and Neil Taylor was introduced to the fray.
Lita headed wide from an O'Neil corner and McDonald lashed a volley over as the home fans became ever more disgruntled.
Rangel's volley was palmed away by Steele and Pratley headed over before Sinclair came to the rescue, finding space as he cut into the box from the left flank and getting away a right-footed shot that took a deflection to creep inside the near post.