A series of other chances went begging too as Nigel Clough's side, who had defender Jay McEveley sent off at the death for a second bookable offence, hung on for dear life, and it took a fine reaction save from Steve Harper to keep out DJ Campbell's rare first-half effort for the visitors.
With West Brom having won at Scunthorpe earlier in the day, another bumper crowd of 47,505 left the ground reflecting on another two precious points which had slipped away after Saturday's 2-2 draw at Sheffield Wednesday.
After surviving an early scare when Jose Enrique's slip allowed striker Campbell to run at the home defence, the Magpies might have been in front within two minutes.
Kevin Nolan's ball down the line was collected by Peter Lovenkrands, who picked out strike-partner Andy Carroll at the far post, but he directed his downward header well wide of the target.
Rams defender Jake Buxton steered a hurried fourth-minute clearance just wide of his own goal, and the visitors were spared only by the crossbar seven minutes later as Ryan Taylor's free-kick left Bywater rooted to the spot.
Carroll headed a 15th-minute Enrique cross wide and Lovenkrands volleyed across the face of goal seconds later with the home side dominating.
Bywater pulled off a vital one-on-one save to deny Ryan Taylor as he ran on to Jonas Gutierrez's superb reverse pass three minutes later, but opposite number Harper had to match his feat when Campbell turned Stephen Pearson's 25th-minute cross towards goal at the near post.
But for all Newcastle's pressure, they simply could not make the breakthrough and Derby somehow got to half-time on level terms.
The visitors started brightly after the break with skipper Robbie Savage firing over from distance with 49 minutes gone, and with Carroll getting little change out of central defender Miles Addison, the Magpies' efforts became increasingly laboured.
They might have gone ahead with 57 minutes gone when Carroll crossed for defender Steven Taylor, but he headed straight at Bywater.
The keeper got down well to keep out Alan Smith's long-range effort seconds later as Chris Hughton's men started to turn the screw once again, but too often, they were let down by wayward passes.
Hughton sent on Shola Ameobi for Carroll, whose battle with Addison had become increasingly physical, with 25 minutes remaining.
But it was fellow newcomer Ranger who went desperately close with three minutes remaining, volleying Nolan's cross against the underside of the bar only to see his shot bounce on the line and drop into Bywater's arms.