Ademola Lookman, with his first goal in 394 days, and Bernard put the hosts ahead inside 14 minutes but they struggled once Michael Bostwick pulled one back.
The 58-place gap between the two sides was barely evident in the hour's football which followed as an Everton team lacking in confidence after just one win in their previous eight matches struggled to find the necessary invention to make the game safe.
It made for a tense affair with the threat of an equaliser from Lincoln, who two years ago became the first non-league club since 1914 to reach the quarter-finals, adversely affecting the Premier League side, who made it past the third round for only the second time in five years.
Silva made five changes, handing Leighton Baines the captaincy on only his fifth start of the season, while also finding a place on the bench for midfielder James McCarthy for the first time since since his horrific double-fracture of his right leg almost 12 months ago.
Lookman signalled his statement of intent within 57 seconds, cutting in from the right and firing a left-footed shot just over, but his long wait for a first goal since December 2017 ended when he converted a far-post header from Baines' cross.
When Gylfi Sigurdsson and Dominic Calvert-Lewin combined for Bernard to cleverly dink over Josh Vickers, back in the Lincoln goal for the first time since Boxing Day after injury, Everton already had one foot in the fourth round.
But it was the last real piece of quality to come from the hosts with Lookman's curling 22nd-minute shot just wide their final attacking threat.
Lincoln, having overcome their early nerves, began to play more football and with their big front man John Akinde inducing unnecessary panic in centre-backs Kurt Zouma and Yerry Mina there were plenty of free-kicks around the penalty area to take advantage of.
They duly did in the 28th minute when Jordan Pickford saved Jason Shackell's header only for Bostwick to follow up from close range, scoring against Premier League opposition in this competition for the second time after finding the net against Newcastle in 2011.
Momentum switched in an instant as Everton lost all composure while their visitors' confidence visibly grew and half-time came at the wrong time for them.
Silva sent on midfielder Andre Gomes and striker Cenk Tosun for Tom Davies and Calvert-Lewin for the second half.
Sigurdsson's 30-yard drive was batted away by Vickers but still Everton did not look like they were in full control and although they dominated possession a lack of cutting edge always offered Lincoln hope.
Visiting hopes were raised when Toffees top scorer Richarlison, sent on as an attacking change 10 minutes from time, had to head out from under his own crossbar to deny Shackell at a corner.
But their hopes of snatching a replay disappeared when substitute Shay McCartan had a penalty claim rejected after going down under the combined challenge of Zouma and Mina.