Northampton began well with on-loan signing Joel Byrom going close on his debut with a shot from the edge of the box which went wide, while Kaid Mohamed saw his low strike from 18 yards saved by Jayson Leutwiler.
Shrewsbury were reduced to 10 men in the 27th minute when Liam Lawrence was shown a straight red card by referee Darren Deadman for a challenge on Byrom.
Marc Richards went close with a header, Byrom had an effort blocked and Ivan Toney was denied by Leutwiler as the Cobblers looked to take advantage of their one-man superiority, while Shrewsbury offered little in the final third.
But, having managed to reach half-time on level terms, Shrewsbury carved out their first openings after the restart. James Collins was denied by Matt Duke and then had an effort blocked, while Jordan Clark rattled the bar from the edge of the box just past the hour mark.
And it was the visitors who broke the deadlock in the 63rd minute when Collins found space in the box to fire into the top corner from 12 yards.
Northampton never really recovered from going behind and did not look like getting back on level terms until Sinclair grabbed the equaliser in the last minute when he headed home a cross from another substitute Lawson D'ath.