The 19-year-old hit an unbeaten 34 off 10 balls as his side posted 174-7 after being put in to bat at Eden Gardens.
Kolkata looked to be cruising in their chase before spinner Rashid claimed 3-19 in his four overs.
The Afghan was also involved in the run out of Nitish Rana and took two catches in the final over of the match.
Two-time champions Kolkata could only manage 161-9 in their chase in front of a packed home crowd.
Victory means Hyderabad will go to Mumbai for the final to face Chennai Super Kings, who beat them in the qualifier at the same venue on Tuesday, and who are led by former India captain MS Dhoni.