Tom Hamer's first-half header was enough to seal three points as City matched their club record unbeaten run.
Hamer's header - from an Adam Reach free-kick delivery - settled the match and was a 19th consecutive league game without defeat.
Head coach Michael Skubala made three changes from Saturday's excellent win at Cardiff City, with Ryan Oné coming into the side for his second City start, while Tom Bayliss and Jack Moylan also returned to the starting XI.
The Imps, who had stayed in the south-west to aid their recovery after a physically-demanding weekend in South Wales, started brightly and created a chance inside the opening minute when Tendayi Darikwa swung in a cross from the right which found its way through to Rob Street. He shot over, but it was an encouraging start.
The early chances continued to go the way of the visitors, albeit without too much threat for the home goalkeeper Joe Whitworth. Hamer's first long throw of the match, after six minutes, found the perfectly-timed run of Conor McGrandles - but the Scottish midfielder could not get the ball out of his feet and settle for a corner.
When the set-piece was delivered by Adam Reach it found Sonny Bradley, but his header back into the area did not find a team-mate. Moments later Bayliss picked up a loose ball from outside the area and let fly with a right-footed shot which went over.
A free-kick, awarded for a foul on Rob Street by Liam Oakes, provided an opportunity from a central position. Bayliss dummied a shot before Moylan rolled the ball for Reach whose shot was blocked by the wall.
A couple of Imps corners came to nothing, while Moylan got a shot away - from an early pass from McGrandles - after 20 minutes but it was low and wide.
Moylan created an opportunity again after 25 minutes, this time collecting the ball from Bayliss who had picked the ball up close to the centre circle. Moylan, faced with three home defenders in front of him, fashioned a shooting opportunity but fired just wide.
With the game approaching the half-hour mark, the home side had their first spell of sustained pressure. George Wickens was out quickly to get to Josh Magennis' cross, which was looking for in-form Jayden Wareham, after 26 minutes then held a shot from Magennis which was straight at him.
Wickens was required for a big save to punch away a shot from Jack Aitchison and keep the score goalless.
That proved decisive two minutes later when the Imps took the lead via a set-piece. Moylan was fouled in a wide left position, providing a perfect opportunity for Reach to deliver a ball into the area where Hamer stooped to head the ball into the bottom left corner.
City looked for a second goal before the interval, with a powerful Reach effort blocked by a combination of Ilmari Niskanen and Pierce Sweeney after Hamer's throw from the right found its way through to him.
And Whitworh did well to get to the ball ahead of Oné and, shortly afterwards, save a shot from the same man with his legs after Moylan slid him in.
There were no changes from either side at the break, with the Imps again starting brightly as Moylan flashed an effort wide two minutes after the restart when Reach's corner was played back into the area by Bayliss.
And Bayliss led a counter moments later before feeding Oné, whose shot was palmed away well by Whitworth. Moylan's effort on the rebound was blocked by Charlie Cummings.
New Grecians manager Matt Taylor had seen enough with a treble change 10 minutes into the second period - sending on Danny Andrew, Akeel Higgins and Reece Cole.
House connected well with a header when Moylan's cross was headed back into his path, with 58 minutes on the clock, but understandably the visitors then had to stand up to plenty of pressure from the home side. As usual, City had no issue allowing the Grecians more possession but did rather limit their chances.
A deep cross, with 20 minutes to go, found Magennis but he could not get a clean connection on his header - under pressure from Hamer - and he nodded behind.
Skubala made a treble change of his own, sending on Dom Jefferies, Ryley Towler and Reeco Hackett.
Wareham flicked a header over after 75 minutes, from a deep Sweeney cross, and the No.9 fired wide after 85 minutes when he threw himself at Niskanen's deliver from the right.
Higgins saw a shot blocked by a flurry of home players right as the game ticked over into time added on.
And although the home side won a couple of later corners, some real determination from the Imps ensured they held on for a fifth straight win and clean sheet.
Exeter: Whitworth, Woodhouse, Sweeney, Cummins (Mendes Gomes, 82), Oakes (Higgins, 56), Brierley, McMillan (Andrew, 55), Niskanen, Aitchison (Cole, 56), Magennis, Wareham. Subs not used: Bycroft, Cox, McDonald.
Imps: Wickens, Darikwa, Reach (Elerewe, 89), Bayliss, Moylan (Towler, 70), McGrandles, Bradley, Street (Lloyd, 82), House (Jefferies, 70), Onè (Hackett, 70), Hamer. Subs: Jeacock, Varfolomeev.
Referee: Carl Brook
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