Kings XI???s victory hopes seemed all but over as they slumped to 99-6 in reply to their Hyderabad???s 185-5, which featured 81 off 52 balls from skipper David Warner.
But Miller took the match to the last over, smashing nine sixes in an unbeaten innings of 89 off 49 balls before his side finished just short on 180-7 off their allocated 20 overs.
Earlier, Warner cashed in on anything short after winning the toss as he and Shikhar Dhawan put on 46 runs in the opening four overs.
Dhawan struck 24 before he was stumped by a distance off Glenn Maxwell???s first ball but there was no containing Warner, who became the top run-scorer in the tournament in the 10th over before bringing up his fifty and his team???s hundred with a six in the 13th over.
Moses Henriques (28 off 24) and Eoin Morgan ??? who hit his second and third ball from Maxwell for six in a knock of 17 off seven ??? both kept the run-rate up before a six apiece from Lokesh Rahul and Karn Sharma in the final two overs completed the onslaught.
In reply, Murali Vijay flicked Trent Boult???s first ball for six as the left armer???s first over went for 15.
However, the opener could not build on his strong start thanks to a great catch from Dhawan, who did well to cling onto a tough chance low to his right at cover.
Bipul Sharma grabbed a second wicket when Manan Vohra holed out at long-on and when Maxwell was undone by the extra pace of Boult, the chase was faltering at 60-3.
Another great catch ??? this time from Rahul, running in from mid-off ??? removed Saha and the game seemed as good as up.
But Miller had other ideas and plundered two sixes off the 18th over, bowled by Boult, to keep his side???s hopes alive.
The first two balls of the final over bowled by Ishant Sharma also disappeared over the rope but with 28 needed off the final six deliveries, the task proved just too tough.