Exeter twice led on home soil, but a 73rd-minute equaliser from Brad Smith has set up a replay at Anfield in what will be a money-spinning affair for the Grecians.
Liverpool went straight down the other end and had a half-chance when the ball dropped to Smith inside the box, but his effort was blocked en route to goal. The Reds were level in the 12th minute, however, when Sinclair converted from inside the box after some slack defending from the hosts.
Joao Teixeira was next to come close for Liverpool, but the midfielder's strike was deflected wide of the post by Jordan Tillson, before Cameron Brannagan brought a smart save from the home side's goalkeeper Robert Olejnik.
Exeter had two more chances late in the first half through David Noble and Christian Ribeiro, but both missed from close range. The League Two side were not to be denied, however, and regained the lead in the first minute of stoppage time at the end of the first period when Lee Holmes's corner drifted over Adam Bogdan's head.
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The home side had a shout for a penalty early in the second period when Ribeiro went down inside the box, but replays showed that the defender had slipped.
Benteke must have thought that he had scored in the 51st minute when he headed towards goal from close range, but Olejnik was on hand to make the save and keep the League Two team on course for the fourth round.
Ribeiro remained a threat down the right for Exeter and came close with another header just before the hour, but his attempt dropped wide of the post, before scorer of the game's first goal, Nichols, curled one over the crossbar from outside the box.
It was the home team that continued to look the most likely as the second half developed, with Nichols firing another free kick over the crossbar during a period where Liverpool were struggling to keep hold of the possession.
The visitors managed to level the scores in the 73rd minute, however, when Australian international Smith converted from close range after a low Sheyi Ojo cross had caused problems inside the Exeter boss.
Teixeira came close to a third for Liverpool in the 80th minute after breaking into the box, but his low strike was kept out by the Exeter stopper, who had a strong game between the sticks. It was the visitors that looked the team most likely to win it late on, but Exeter held on to force a replay at Anfield.