After being put in to bat, Hyderabad posted a formidable 194-9 on the back of Abhishek's 59 off 22 and veteran South Africa international Heinrich Klaasen's 59 off 39.
In reply, five-time winners Chennai lost opener and India's T20 World Cup hero Sanju Samson cheaply for 7 off 3.
Impact substitute Ayush Mhatre hit a quickfire 30 off 13, giving the chase momentum, but a sharp catch by Klaasen cut short the teenager's heroics.
A flurry of quick wickets followed, stymying the run flow in the middle overs.
Despite handy contributions by Australian T20 specialist Matthew Short and Indian batter Sarfaraz Khan, Chennai's innings never recovered from the early blows.
Chennai skipper Ruturaj Gaikwad blamed the defeat on the lack of partnerships in the middle over.
"It was just about 80 runs needed in (the last) 10 overs in the chase," he said.
"From there it was just about building a couple of partnerships."
Malinga was ably assisted by Indian all-rounder Nitish Kumar Reddy, who returned figures of 2-31 in his four overs.
Sunrisers captain Ishan Kishan was all praise for his bowlers who he said helped his team cross the line despite being "20-30 (runs) short".
"Lovely when you have a bunch of young bowlers coming up with their plans," said Kishan.
Earlier, Sunrisers Hyderabad got off to a flying start with openers Abhishek and Travis Head notching up 75 runs in the powerplay.
Head was the first to go, immediately followed by captain Ishan Kishan, who was out for a first-ball duck.
Abhishek followed soon after.
Klaasen held steady at one end before being bowled by pacer Anshul Kamboj.
For Chennai, Kamboj returned the most productive figures, taking three wickets for 22 runs.
One of the most successful franchises in the T20 tournament, Chennai are still waiting for the return of India legend M.S. Dhoni, who is recovering from a calf injury and is yet to play a game in this edition.
