Neither side were able to gain an early foothold in the game played on a slick surface battered by hours of rain.
However, Shrewsbury were first to threaten when a corner was swung into the path of Bryn Morris, whose left-footed half-volley clipped the top of the Imps' bar.
Moments later, Rhead was shown a yellow card for cynically clattering into Shrewsbury goalkeeper Henderson, who had to be given treatment on the pitch for several minutes.
The deadlock was broken just after the quarter-hour mark as an inswinging corner was spilt by Henderson into the path of Whitehouse, who made no mistake from close range.
The half-time whistle blew on a narrow Lincoln lead as Shrewsbury were left feeling aggrieved at a number of decisions seemingly not going their way.
Shrewsbury grew increasingly frustrated as the game went on, however, as they failed to take their chances.
Indeed, Lincoln could have doubled their lead on 80 minutes when Whitehouse attempted to volley a high delivery goalwards, but it failed to hit the target.
In the end, the game would be settled by the one goal as Danny Cowley's charges became the fifth fourth-tier side to win the trophy.
Shrewsbury, meanwhile, have now lost each of their four appearances at the national stadium.
LINCOLN CITY (4-3-3): Allsop; Eardley, Bostwick, Waterfall, Habergham; Whitehouse, Woodyard, Frecklington; Rowe (Anderson 64'), Rhead (Palmer 63'), Green (Long 90+5')
SHREWSBURY TOWN (4-1-4-1): Henderson; Bolton, Nsiala, Sadler, Beckles; Morris (Rodman 67'); Whalley (Gnahoua 86'), Nolan, Godfrey, Thomas (Payne 75'); Morris