Yevkurov was injured when his motorcade was hit by a suspected suicide car bomb near the city of Nazran, the volatile southern republic's Interior Ministry said.
Ingushetia's Interior Ministry spokesman Kaloi Akhilgov said the president was in stable condition following an emergency operation. Yevkurov's driver died at the scene of the assassination attempt.
Russian President Dmitry Medvedev has ordered the federal Interior Ministry and the FSB security service to carry out an investigation into the blast, and has also sent his envoy to the region, the Kremlin said.
Medvedev qualified the blast as a terrorist attack linked to Yevkoyev's efforts to end a spiral of violence in the republic.
Yevkurov, a former military intelligence official, became president of the republic last October. In a recent RIA Novosti interview, he said he would offer an amnesty to remaining militants, but pledged "to eliminate" those who would not lay down their arms.
Ingushetia, which borders on Chechnya, witnesses frequent militant attacks on officials and police.
Russia's North Caucasus republics have been swept by a wave of violence recently, two months after the government formally ended its decade-long counterterrorism operation in Chechnya, which witnessed two brutal separatist wars in the 1990s and early 2000s.