Nehra's opening burst helped restrict Afghanistan to 115-8 in its major tournament debut.
Opener Noor Ali topscored with an assured 50 while Asghar Stanikzai scored 30.
Vijay anchored India's run chase and the 2007 champion got home with 31 balls to spare at 116-3.
The Afghans were in early trouble after being sent in.
Man of the match Nehra claimed Karim Sadiq (0) and Mohammad Shahzad (6) cheaply as they deflected catches to the wicketkeeper from short deliveries.
Afghan skipper Nowroz Mangal (5) soon lofted Ravindra Jadeja's left-arm spin to mid-off and India was on top at 29-3 in the sixth over.
But Ali and Stanikzai lifted their side with a fourth-wicket stand of 68 off 66 balls until their exits set off a late slump.
Ali's innings spanned 48 deliveries and included four fours while Stanikzai smacked three sixes.
India's pursuit was paced by the right-handed Vijay, who struck two fours and three sixes off 46 balls.
Even though they lost left-hander Gautam Gambhir (4) and Suresh Raina (18) cheaply, India was never threatened.
Captain Mahendra Singh Dhoni smashed two sixes in an unbeaten 15 off six balls to ensure India won with 5.1 overs to spare.
Left-hander Yuvraj Singh contributed 23 not out off 22 balls.
Both teams still have to face South Africa in Group C.
