Grant's moment of misfortune means Leicester have won their opening two away games of the season for the first time, and Nigel Pearson's men were good value in front of a 23,000-plus crowd.
It was a shock for Derby, who have started the season well, and who in fact set off well, with England Under-21 midfielder Will Hughes impressing.
He linked played up well and, after just two minutes, was involved in a move which saw Jonny Russell claim he was bundled over on the edge of the box.
Nothing was given but Derby piled forward, with Craig Bryson firing over in the fifth-minute, before Leicester started to threaten through Chris Wood.
Chris Martin then hooked a shot over for Derby, before Jamie Vardy went close for Leicester with 17 minutes gone - the striker finding Grant in his way as he drilled in an effort.
Bryson then led the charge for Derby and Matty James hacked him down to halt a move, before Craig Forsyth got in and was kept out by Kasper Schmeichel.
Derby were edging it in terms of possession but in the 37th minute, Leicester scored.
Liam Moore headed down a corner and, after Vardy's flick was hacked off the line by Paul Coutts, an unsuspecting Grant was hit and the ball went in off him.
Leicester were in again just before half-time but a great tackle by Richard Keogh robbed Wood of possession.
Hughes and Bryson had long-range efforts go away from the target as Derby struggled to break down a solid Leicester defence, but the home side were close to equalising in the 67th minute.
Russell turned Ritchie de Laet on the left and whipped over a cross which ran to the back post where Kieron Freeman drove in a lot shot which Schmeichel blocked.
Then the visitors almost scored a second in the 73rd minute when De Laet surged forward and put Vardy in on the right, but the striker's low shot was blocked by Grant.
Derby came again when Hughes found space on the left but shot wide under pressure, and the midfielder then set-up substitute Jamie Ward in the 82nd minute but his shot was charged down.
Keogh also flashed a header wide, but Derby lacked bit and Leicester comfortably held on to take the points and end Derby's unbeaten start to the campaign.