The Leeds star opened the scoring as his menacing free-kick from the right bounced into the corner of the net.
As he celebrated the goal, Raphinha removed his shirt to pay his respects to Marilia Mendonca, who died in a plane crash on Friday.
Fellow Brazilian Neymar also paid tribute to Mendonca during Paris Saint-Germain's 3-2 victory at Bordeaux on Saturday, a match that saw Neymar net twice.
But just 73 seconds after the Whites went ahead they were pegged back by a stunning Harvey Barnes goal.
Barnes collected the ball on the left before striking with incredible precision into the top left-hand corner as both sides shared the spoils.
In a breathless opening ten minutes, Leeds twice went close to opening the scoring and had strong appeals for a penalty turned down after Barnes had been first to threaten for Leicester.
Barnes' effort was easily gathered by Illan Meslier and at the other end Jack Harrison's curling shot was turned away at full stretch by Kasper Schmeichel.
Schmeichel then produced a point-blank save to keep out Kalvin Phillips' header from Raphinha's corner and referee Darren England was unmoved when the Brazilian winger went tumbling under Ricardo Pereira's challenge.
There was no let up as Leeds' collective energy levels appeared to have been restored, with Raphinha and Dan James both threatening.
Raphinha curled Leeds into a deserved 26th-minute lead when his free-kick bounced inside Schmeichel's far post, but the home side's advantage was short-lived.
Straight from the restart, Barnes cut inside from the left and after shifting the ball on to his right foot, curled a brilliant equaliser beyond Meslier and into the top corner.
Youri Tielemans flashed a shot wide soon after the restart, but Leeds swarmed back on to the offensive.
James' low cross was blocked, Caglar Soyuncu headed inches wide of his own post and Harrison somehow managed to miss in front of an open goal at the far post following Phillips' header.
James fired another effort narrowly wide and Rodrigo miscued a volley at the far post as Leeds reproduced the high-octane form that had won them so many admirers last season.
Having failed to convert several chances, Leeds were then given a let-off in the 67th minute when Ademola Lookman's far-post effort was ruled out for offside by VAR.
To have gone behind would have been harsh on Leeds but after more magic from Raphinha on the right, their final pass continued to elude them.
James spurned another scoring chance after Rodrigo had regained possession and, as Leeds continued to live dangerously at the back, Leicester defender Soyuncu dragged his effort wide.
Wilfred Ndidi scuffed his shot as Leicester still threatened to snatch all three points in the closing stages before Leeds' hopes of victory ended when Raphinha's thumping drive whistled over.