Cavonnier had inched his way into a fractional lead over the favourite, Unbridled's Song, with Grindstone a length and a half back in fourth, as the field turned for home.
Unbridled's Song faded and Cavonnier and his rider Chris McCarron seized a two length advantage, but Grindstone came with a powerful late run to pip Cavonnier in the last stride. It was the closest finish since Tomy Lee edged out Sword Dancer by the same margin in 1959.
Prince Of Thieves, also Lukas-trained and bumped at the start was a further three and a half lengths back in third, a neck and a nose in front of Halo Sunshine and Unbridled's Song.
Lukas has now won the last six Triple Crown races - a record - and joins James Fitzsimmons and Max Hirsch as the only trainers to have the Kentucky Derby three times. Grindstone, a 5.9-1 shot, now goes for the Preakness Stakes, the second leg of the American Triple Crown.