The striker was heckled by the fan after taking a short corner in the final seconds instead of knocking the ball directly into the penalty area.
Drogba reacted furiously and at the final whistle was still engaging the spectator in an ugly argument as the rest of the players shook hands.
Saving grace: Stoke keeper Asmir Begovic is beaten by a header from Drogba
The Ivory Coast forward allegedly told the fan to repeat what he had said, but this time to say it directly to him.
It took the intervention of Paulo Ferreira to defuse the situation by dragging a still furious Drogba away from the scene and down the tunnel.
The incident was indicative of the frustration Drogba is clearly still feeling after an indifferent season, in spite of scoring his first goal for 10 weeks to salvage a point for Chelsea.
If lifting the Barclays Premier League trophy at their Stamford Bridge premises this season now seems beyond Chelsea, the Champions League is another matter.
VIDEO: Watch Drogba confront supporter
Wednesday night against Manchester United will be as night is to day compared to the ferocity of Saturday's humdinger at the Britannia Stadium, yet the signs were there that Chelsea are comfortable in their own skin again.
Perhaps no other side, including United, could have withstood the second-half barrage of pressure which Stoke mounted, but Chelsea did not wilt.
United will be different, of course.
Yet, the hunger Chelsea will need to display will be no different.
And their forwards will need to be as trusting of each other, just as Drogba and Anelka, reunited after five Premier League matches apart, were on Saturday.
Carlo Ancelotti will consider turning back to Fernando Torres, yet the mood among Chelsea fans was that he should resist that urge.
CHELSEA'S NOT-SO-FAB FOUR In their last nine matches in all competitions, Chelsea have managed just 11 goals.
Three have come from defenders, five from midfielders and three from strikers.
A strike-force with a combined value of nearly ?100m managed just one shot on target at Stoke.
Drogba blazed wide six times, while Chelsea's other three forwards mustered only one shot between them.
Ivan Speck looks at their woes and assesses their form.
Out of form: Drogba (left) grabbed his first goal since January seemed reassured by Anelka (right)
Saturday's goal was his first since January. He seemed reassured by the presence of Anelka alongside him once more.
Perhaps his ego requires a partner who is prepared to play the subservient role.
At his best he is virtually unplayable, but his best requires him to be fully restored to health following his autumn bout of malaria.
The sharpness is back and so, too, appears to be the hunger.
Could be about to explode into irresistible life.
Played 90mins Goals 1 On target 1 Off target 6.
The last Chelsea striker to score - 429 playing minutes before Drogba's equaliser at Stoke - but his rounded play is ofter under-appreciated.
Proof could be seen in the unselfish manner in which he dropped deep to curl a succession of inviting crosses into Drogba's path at the Britannia Stadium.
Seems to be relishing the freedom of movement which the reinstalled 4-4-2 formation gives him.
His self-confidence runs deep, but he is more reliant than Drogba on good service to provide his chances.
Played 61mins Goals 0 On target 0 Off target 1
Unfit and fitful: Torres (left) has been hampered by injury and Kalou (right) lacks confidence
Two months into his ?50m move from Liverpool, the Spaniard has yet to score his first goal.
His confidence, sapped initially by a series of muscle injuries which robbed him of his lightning pace, has now been absent for two full seasons.
No longer appears capable of racing clear on to through balls, which means he is facing defenders, not escaping from them.
Appeared lost as a second-half substitute at Stoke, running head down into defenders for his half-hour on the pitch.
Played 29mins Goals 0 On target 0 Off target 0
Has been a fitful presence in the side since joining in 2006. In five seasons at Stamford Bridge he has scored just 52 goals in 221 appearances, 10 of them this campaign and few of them crucial ones.
A cool finisher when he bursts into the penalty area unchecked, he rarely attempts shots from distance, meaning that opposing defenders can stand off more and block his pass routes.
Being a substitute so often robs him of continuity and consequently confidence in his cameo appearances.
Played 29mins Goals 0 On target 0 Off target 0
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Explore more:People: Didier Drogba, Fernando Torres, Nicolas Anelka, Carlo Ancelotti Places: Liverpool