Sinclair doesn't score many, his last goal came way back in 2000, but this was an important one and condemned Rotherham to a 21st league defeat of the season.
The former Leicester defender bravely got his head to a cross from John Oster as he headed past Mike Pollitt via the underside of the crossbar in the 14th minute.
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But any thoughts Rotherham might have harboured of securing a fourth win of the season were quickly dashed.
Poor defending contributed to Burnley's equaliser. Tony Grant was allowed too much time and space just eight yards out after a cross from Mo Camara had been flicked on by Dean Bowditch, a young striker on loan from Ipswich.
Rotherham must have feared the worst when Sinclair's header put the home side in front but they showed tremendous resiliance and might have equalised several times.
Will Hoskins was denied by a good block by Jensen before a free-kick from Paul Hurst bounced off a post.
Rotherham continued to be the better side and there was little to cheer for Burnley's lowest league crowd of the season, just 10,589.
Millers midfielder Paulo Vernazza twice went close from distance while at the other end Pollitt had to look lively to tip over a header from Bowditch.
The visitors, spirited throughout, were denied an injury-time equaliser when Jensen dived away to his left to keep out a shot from Gilchrist who had obviously started to fancy himself as a scoring threat.