Despite a strong performance, Everton managed to gain just a point against Brimingham City in a result that leaves them two points above the relegation zone.
Sebastian Larsson impressively cancelled out Diniyar Bilyaletdinov's goal to keep the pressure on David Moyes' injury-ravaged squad.
A statistic of two wins from nine league games at Goodison lies at the root of Everton's problems and, while the pending return of injured players and a potential run in the Europa League point to brighter times in the new year, draws are not worth much in current Premier League currency.
"I am concerned because we are at the wrong end of the table but I've been concerned for a couple of months," said Moyes. "We can't be waiting for people to come back, we need to win games now."
Everton might have cemented victory in the opening 20 minutes, during which Bilyaletdinov put his team in front after ghosting around three defenders to collect Louis Saha's touch and finishing confidently. Minutes later Saha had a goal ruled out for offside despite television replays showing him to have been well onside when he collected Tim Cahill's pass.
It only appeared a matter of time before the home side put the game beyond Birmingham's reach, but the confidence and resilience City have gathered during a five-match winning run was evident and, on their first attack of note, Larsson and Christian Benitez were allowed the space to exchange passes before the former finished expertly from inside the area.
Birmingham were required to defend manfully in the second half although their manager, Alex McLeish, was honest enough to admit that not only was Saha's goal onside but that Carr also handled with Saha threatening late in the game.
Still, McLeish could not disguise his satisfaction at seeing his team play below recent par but record a ninth consecutive unbeaten game. "We were hoping to keep our unbeaten run intact and that's a great way to enter Christmas," he said.
"Our primary target is to keep our Premier League status and, once we can achieve that – and we certainly can achieve that – then we will look at anything else as a bonus."
Everton (4-1-3-1-1): Howard; Hibbert, Neill, Heitinga, Baines; Fellaini; Bilyaletdinov (Yakubu 70), Osman (Vaughan 90), Pienaar; Cahill; Saha. Substitues not used: Nash, Coleman, Duffy, Agard, Baxter.
Birmingham (4-4-2): Hart; Carr, R Johnson, Dann, Ridgewell; Larsson, Bowyer, Ferguson, McFadden (Fahey 69); Benitez (Vignal 82), Jerome. Substitues not used: Taylor, Phillips, McSheffrey, D Johnson, Carsley.
Referee: S Attwell
Booked: Birmingham: Larsson, Hart.
Man of match: R Johnson