Chasing a stiff 206, Morkel blasted three sixes and two fours off the penultimate over bowled by Virat Kohli as the hosts overhauled the target in the last ball to record the second highest runchase in IPL history.
Faf du plessis (71) and skipper Mahendra Singh Dhoni (41) were the other notable scorers for Chennai who have now got their campaign back on track after two successive defeats.
The home team turned the complexion of the game in the 19th over with Morkel racking up 28 runs and needed 15 runs in the last over.
Morkel perished in the second ball of the last over bowled by Vinay Kumar but Dwayne Bravo (25) did the star turn by taking the team to the victory in the tense dying moments of the cliffhanger.
With two runs needed in the last ball, Ravindra Jadega edged the ball to the boundary to send the home crowd in a frenzy and trigger wild celebrations in the Chennai dug-out.
It was a remarkable come from behind victory for Chennai, who looked down and out at one stage but RCB skipper Daniel Vettori's gamble to give Kohli the 19th over backfired in a big way.
Earlier, Chris Gayle hit a blistering 68 and shared a 109-run stand with Kohli (57) to power Royal Challengers Bangalore to a mammoth 205 for eight. But that proved insufficient in the end.