Karnataka home minister KJ George told NDTV it was a terror attack, although no group has yet claimed responsibility for it.
The blast took place outside the popular Coconut Grove restaurant on Church Street off MG Road, just after 8.30 pm.
Police sources said it was a low-intensity blast from an Improvised Explosive Device, which was filled with shrapnel, used ammonium nitrate and was fitted with a timer device. The bomb had apparently been placed in a drain on the roadside. The bomb disposal squad and a forensics team have been rushed to the spot.
The woman, Bhavani, 38, a resident of Tamil Nadu, had sustained fatal injuries to her head from shrapnel in the bomb. She died soon after being taken to the hospital. The man, Karthik, her relative, suffered minor injuries and is better now, said City Police Commissioner M N Reddi.
Union home minister Rajnath Singh said he has spoken to Chief Minister Siddaramaiah and the Centre is ready to offer all help.
The Chief Minister visited the blast site late on Sunday night and met the injured victim at the hospital.
Spoke to Karnataka CM Shri S Siddaramaiah regarding the Bangalore blasts. He apprised me of the situation. Centre ready to provide all help.; Rajnath Singh (@BJPRajnathSingh) December 28, 2014
Union law minister Sadananda Gowda, who represents Bengaluru North in Parliament, has rushed to the blast site. "I request fellow Bengaluruians not to panic. We r with u (sic)," he tweeted.
The police, which also requested for calm, said the situation was under control. Additional police forces have been mobilised in the city, where the threat perception had been high in view of the year-end festivities.
Earlier this month, Mehdi Masroor Biswas, a young engineer from Bengal who was working in the city, was arrested for allegedly running @shammiwitness, a pro-Islamic State twitter handle. Sources told NDTV they would investigate all angles.