Doyle had gifted NK Maribor the half-time lead in their Group H clash when he swung at and missed Jonathan Spector's backpass to allow Dalibor Volas to score a comical opener.
But Scotland winger Chris Burke levelled and then Elliott, a recent signing from Burnley, earned Chris Hughton's side a memorable victory with a classy strike to give his side real hope of making a push for the knock-out stages.
City boss Chris Hughton made several changes for his side's first away group fixture, including the decision to field Doyle in place of regular starter Boaz Myhill.
Maribor coach Darko Milanic, who masterminded Rangers' downfall at the play-off stage, fielded Ibrox goalscorer Volas up front at the Stadion Ljudski.
Both sides were seeking their first points having lost their respective openers but the onus was on the Slovenian champions to take all three on home soil.
Marlon King's early header was held by Jasmin Handanovic and stand-in captain Liam Ridgewell's header just lacked enough power.
Maribor went close when captain Marcos dragged his shot just wide and Doyle then showed superb reflexes to tip away Goran Cvijanovic's point-blank header.
At the other end King had two half-chances either side of Keith Fahey's off-target volley.
When the deadlock was broken, it was in farcical circumstances.
Doyle, so impressive moments earlier, somehow missed Spector's backpass and then seemed to watch helplessly as the ball rolled behind him and Volas gleefully put the ball in the net.
Three Birmingham half-chances came and went in one move as King's shot was blocked, Burke's follow-up was saved and Jean Beausejour's attempt went wide.
The goal briefly quietened the travelling contingent of just over 600 Birmingham fans and Ales Mertelj threatened to shut them up completely when his searing drive from distance whistled over the bar early in the second half.
Ridgewell, booked just before the break, was perhaps unwise to tangle with Dejan Mezga on the touchline while attempting to take a throw-in but referee Tom Harald Hagen decided no action was necessary.
Hughton had strikers Nikola Zigic and Chris Wood stripped and ready to come on when the equaliser came in the 64th minute.
King held the ball up well and flicked it on to Burke, who then showed superb technique to poke the ball around his marker and slot it under the onrushing Handanovic.
Zigic immediately replaced King anyway and he caused Maribor immediate trouble.
Elliott could then have done better with a chance soon after but Handanovic came out deny him.
But Birmingham were buoyant and Elliott soon struck the winner.
A speculative punt forward 11 minutes from time just eluded Morgaro Gomis' control, but when the loose ball was not dealt with, Elliott was alert to the opportunity and drove his right-footed half-volley inside Handanovic's near post.
Handanovic could perhaps have shown more strength to keep the shot out but Elliott did not care as he sprinted away to celebrate in front of the joyous Birmingham contingent.