The Championship side recovered from an opening day defeat to Derby in emphatic fashion and were set on the way by first-half goals from Jonathan Howson, Luciano Becchio and Lloyd Sam.
Neil Kilkenny's penalty added to the tally after the break as the Red Imps were condemned to an opening round exit for the fifth successive season.
Summer signing Alex Bruce was handed his competitive debut for Leeds, while the visitors were boosted by centre-back Adam Watts' rapid recovery from an ankle knock picked up in the 2-1 defeat at Rotherham on Saturday.
However, the 22-year-old's inclusion at the heart of defence could not prevent Leeds from breaking the deadlock within the opening two minutes.
Sam jinked inside from the right wing and sprayed the ball out to Sanchez Watt on the opposite flank, with the resulting cross met at the far post by Howson's stooping header.
The League Two side were visibly rattled and left rueing inept defending down the flanks once again five minutes later.
Bradley Johnson pressed forward through the middle and threaded a neat ball to Sam, allowing the 25-year-old winger to whip in a left-wing cross for Becchio to finish from close range.
Only a smart save from Lincoln goalkeeper Joe Anyon moments later prevented Watt from putting his side even further behind, although he could not keep the deficit at two goals for long.
Johnson and Howson came close before Leeds netted their third with Sam heading home Paul Connolly's intelligent dink to the back post in the 30th minute.
After cruising for the rest of the half, United started the second period in the same manner as the game petered out.
However, Leeds remained in control and right-back Paul Green's torrid evening was compounded in the 71st minute when he conceded a penalty for a clattering challenge on Watt, which Kilkenny duly converted.