Lionel Messi opened the scoring from the penalty spot after our attacker was brought down inside the box.
And Alvarez secured the success with his third a fourth goals of the tournament - his first a wonderful solo effort before slotting home his second of the match after the interval.
Their progression to the showpiece event in Qatar is their sixth in total, having won the trophy twice and falling to defeat three times previously.
Lionel Scaloni’s side will learn their opponents following the second semi-final tie between Morocco and France on Wednesday.
The opening 15 minutes were cagey, with neither side creating many clear-cut chances.
But it was Argentina who carved out the first opportunity 20 minutes into proceedings.
Enzo Fernandez collected the ball from 25-yards and powerfully let fly, which tested Dominik Livakovic who dived low to his left and palmed it away.
City’s striker was then in the thick of the action when he beat the offside trap to race into the box, before Croatia’s ‘keeper brought him down after being rounded by La Arana.
Messi stepped up and fired the first goal of the game into the top corner.
Alvarez then stole the show shortly after with a sublime goal to double his side’s advantage.
The 22-year-old picked up the ball from his own half, drove through a number of defenders, entered the box and prodded the into the net.
City’s number 19 was making his sixth consecutive appearance at the World Cup and he was continuing to cause Croatia problems.
He shook off his marker with 30 minutes remaining, utilised his pace to break down the wing before squaring to Fernandez who couldn’t quite reach his pass.
Zlatko Dalic’s side created their best opportunity with just over 25 minutes remaining when an inviting cross into the box was met by Dejan Lovren’s goalbound header, which was blocked and then cleared by Argentina.
It didn’t take long, however, for the South Americans to pull further ahead and it was Alvarez again who tormented the Croatia backline.
Messi received the ball and danced down the right flank before pulling back to our striker who who swept, first time, into the bottom corner.
City’s striker will now have a shot at World Cup glory on Sunday 18 December at the Lusail Stadium.
