Maxwell hammered 95 off 43 balls while Miller made an unbeaten 54 off 37 to propel their team past Super Kings' imposing 205 for four with seven balls to spare at the Sheikh Zayed stadium.
Virender Sehwag (19 off 10) and Cheteshwar Pujara (13 off 10) started Punjab's chase aggressively before departing in quick sucession.
Offie Ravichandran Ashwin, who trapped Pujara in front of the stumps, earned his second lbw by foxing Akshar Patel (2), leaving Punjab at 53 for two in sixth over.
That brought together two of the world's dangerous batsman in the middle - the big hitting Miller from South Africa and the innovative Maxwell of Australia.
The deadly duo did not disapppoint as they took the Super Kings bowlers to the cleaners. Maxwell scored the bulk of the runs in their 63-ball stand and was also at his cheeky best.
The Australian all-arounder enthralled the crowd with his switch hits and hoicks over mid-wicket. Like Miller, who was dropped off Ravindra Jadeja in the ninth over, Maxwell also had the luck on his side. The result could have been different had the Australian not been dropped twice.
His whirlwind innigs comprised 15 fours and a couple of maximus before he missed a Dwayne Smith yorker in the 16th over with Punjab still needig 39 off 26 balls.
Miller got into the groove after Maxwell's fall and got his side closer to the target by smashing Jadeja for two consecutive sixes. With captain George Bailey (17 off 10) at the other end, Punjab eventually strolled to a morale boosting victory.
Earlier, Super Kings kicked off their IPL-7 campaign with a flurry of fours and sixes as Brendon McCullum (67 off 45) and Smith (66 off 43) hit blazing half centuries.
An entertaining 123-run opening-wicket partnership between McCullum and Smith, and a 11-ball 26 cameo by skipper Mahendra Singh Dhoni towards the end ensured Super Kings posted a mammoth score on a good batting surface. The final over saw 18 runs being scored.
The innings started with a bang and the fifth over saw McCullum and Smith stealing 18 runs. The over proved costly for Punjab as Smith hit two consecutive boundaries before Bailey dropped the dangerous McCullum off pacer Parvinder Awana at 22. The Kiwi batsman punished the bowler by hitting a huge six.
McCullum brought up his fifty with a boundary off Rishi Dhawan.
The openers sent a lacklustre Punjab attack on a leather hunt and that helped Super Kings reach 100-run mark in 10th over.
While McCullum was at his dominating best, Smith was steady at the other end, making sure that the loose deliveries did not go unpunished.
It was the slow left-arm orthodox bowler Patel who finally gave Punjab the breakthrough by removing McCullum.
While Smith was looking to continue the momentum after McCullum's dismissal, Suresh Raina began with a boundary.
Medium pacer Laxmipathy Balaji dismissed Smith with Bailey making no mistake at covers this time.
Determined to capitalise on the solid platform laid by the openers, Dhoni launched an all out attack from the very first ball.
Raina's departure after a 19-ball 24 hardly affected the team's rhythm and in able company of Dwayne Bravo, Dhoni managed to take CSK past 200.