Brendan Rodgers' side, for whom defeat would have ended their qualification hopes, had lost back-to-back matches in Group B, including a 2-0 reverse against Leipzig last time out.
And they looked to be heading for a deflating draw when Jean-Kevin Augustin cancelled out Kieran Tierney's first-half opener with 12 minutes remaining at Celtic Park.
However, Edouard hit back within 65 seconds to move the Scottish Premiership champions level with their visitors on six points with two games remaining.
Celtic had rebounded from their Leipzig loss by scoring 13 goals in a three-game winning streak, including a 5-0 thrashing of title rivals Hearts, but they made a timid start to this clash.
The hosts were forced onto the back foot early in the game but broke the deadlock completely against the run of play.
James Forrest's cross eluded everyone except the alert Tierney, who took a touch before sweeping a low strike beyond Yvon Mvogo.
Leipzig knew a victory would secure their spot in the last 32 but they struggled to mount a meaningful response and Craig Gordon remained largely untested.
Mvogo was called into action in the 55th minute, making an untidy stop from Edouard's low shot, but Celtic retained an air of understandable caution.
They were forced out of their shell by Augustin's headed leveller from Konrad Laimer's delivery, but the reaction was rapid as Ryan Christie squared for Edouard to seal a dramatic triumph.
Rodgers – who breathed a sigh of relief when Matheus Cunha hit the crossbar late on – takes his team to pointless Rosenborg on November 29, while Leipzig make the trip to take on a Salzburg side still boasting a 100 per cent record.
Key Opta facts
- Celtic have won three consecutive home games in Europe for the first time since the group stages of the Champions League in 2007-08 (excluding qualifiers).
- RB Leipzig have failed to keep a clean sheet in all eight of their away games in Europe (excluding qualifiers), conceding 16 goals in total.
- Brendan Rodgers is unbeaten in his last eight games as a manager in the Europa League (W6 D2) since losing his very first match in the competition with Liverpool (vs Udinese in Oct 2012).
- Brendan Rodgers defeated a German manager in major European competition for the first time at his seventh attempt, drawing one and losing five of his previous six such games.