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 Blues a memorable victory with a classy strike to give them real hope of making a push for the knock-out stages.
Manager 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, while Maribor coach Darko Milanic, who masterminded Rangers' downfall at the play-off stage, fielded Ibrox goalscorer Volas up front at the Stadion Ljudski.
Marlon King's early header was held by Jasmin Handanovic and stand-in captain Liam Ridgewell's header just lacked enough power before 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.
And when the deadlock was broken, it was in farcical circumstances, as 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.
The goal briefly quietened the travelling contingent of just over 600 Blues 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.
The equaliser came in the 64th minute as 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.
Nikola Zigic replaced King and he caused Maribor immediate trouble, and Blues, buoyed, soon grabbed 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.
Source: PA