The industrious frontman took advantage of a blunder from goalkeeper Tim Howard in the 14th minute and that was enough to give his side victory.
Everton went into the match with high expectations as they a fully fit squad to call on.
Instead they failed to get going and it was only late in the game goalkeeper Paul Robinson was tested.
Blackburn remained solid throughout with Morten Gamst Pedersen and Phil Jones taking control of a crowded midfield.
They almost opened the scoring after two minutes against an Everton side dressed in pink.
Pedersen's clever chip released Martin Olsson but with only goalkeeper Howard to beat he hoisted his shot over the top.
Howard was again called into action four minutes later and dealt comfortably with a header from El-Hadji Diouf following a long ball from Gael Givet.
Blackburn, however, were forced to make a change after nine minutes with Steven N'Zonzi replacing David Dunn, who appeared to have suffered a groin strain.
Sam Allardyce's side then took the lead in the 14th minute following a mistake from Howard.
He raced out his goal to try and clear his lines but instead spilled the ball and Kalinic was on hand to sweep it home.
It was just the boost Blackburn wanted so early in the game but Howard was left distraught.
Everton responded but Michel Salgado made the clearance with Tim Cahill, who had recovered from a slight knee problem, lurking close by.
Blackburn found themselves on the back foot after 29 minutes when Cahill made space in the area only to scuff a shot into goalkeeper Robinson's arms.
Everton were beginning to pick up the pace of the game and Louis Saha got away from Christopher Samba five minutes later. The French striker cut inside only to see his effort blocked.
Then Mikel Arteta's free-kick was deflected before Steven Pienaar tested the goalkeeper, who saved at the second attempt after the ball bobbled up in front of him.
It was now Everton who had the momentum and were beginning to string their passes together.
Sylvain Distin launched a free-kick into the area after 42 minutes and Cahill escaped his marker but his header failed to trouble Robinson.
Phil Jagielka then tried his luck from distance after 57 minutes but his effort went wide of the post.
Two minutes later manager David Moyes switched things around by introducing former Leeds striker Jermaine Beckford for the ineffective Leon Osman.
Allardyce followed suit in the 60th minute by handing Mame Biram Diouf, on loan from Manchester United, his debut. He replaced Kalinic, who had put in a hard shift.
Mame Biram Diouf was looking lively after coming on and saw his shot cleared by Howard, only to be ruled offside.
Then Phil Jones raced forward and skipped into the area but his effort was blocked by Distin.
Play switched down the other end and Saha got into a good position but sent his shot high and wide.
Blackburn nearly extended their lead in the 78th minute when Nzonzi got on the end of a cross from El-Hadji Diouf but headed marginally over the bar.
Everton kept pushing forward and substitute Jack Rodwell made space to get in a header two minutes from time but it went wide of the post.
Then Jagielka thumped in a shot in stoppage time but Robinson was equal to it and prevented the visitors from equalising.