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Shaun Barker's fifth goal of the season gave the Rams a first half lead, but Coventry had a stoppage time goal disallowed for offside, causing a mass protest from the Sky Blues players and staff.
Chris Coleman had demanded that his team came storming out the blocks after several slow starts in recent weeks, and he nearly got the perfect response.
With just 22 seconds on the clock, Clinton Morrison headed the ball down to Jonathan Stead but his low drive flew inches wide.
The attacks continued with Stead going close again in the sixth minute when he latched on to a loose ball and drew a good save from Bywater, before his tenth minute shot was also pushed away.
However, the visitors took the lead with their only effort of the half. Michael Tonge swung in a corner which eventually fell to Barker who sent the ball towards the goal. Keiren Westwood cleared the ball but the linesman indicated the ball had crossed the line and awarded the goal.
The game went flat but the interval brought about a new sense of urgency.
Bywater produced a marvellous block to deny a rocket of a shot from Carl Baker before Derby went up the other end and Rob Hulse dragged his attempt wide.
Coventry tried to force an equaliser but struggled to test Bywater in the Derby goal.
At the other end, Barker thought he'd got his second goal from a corner when he headed beyond Westwood - but referee Andy D'Urso blew for a foul.
Coventry threw bodies forward and they were nearly made to pay when Tonge went on a mazy run and shot towards the bottom corner, but it was just wide of Westwood's post.
The game seemed to be over, but a late surge saw Chris Hussey put in a cross which missed everyone and went in off the post - only for D'Urso to rule out the last-gasp effort in controversial circumstances.