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Miller's dismissal reduced Punjab to 109/4 in 14.4 overs.
In the 13th over Lakshmipathy Balaji struck to get the wicket of danger man Manan Vohra after his 21-ball-31 that had steadied Punjab after the fall of openers.
Vohra was caught by Das at deep cover after a quick-fire knock that was studded with three boundaries and a six.
Kolkata's Rajat Bhatia inflicted second blow to Kings XI Punjab by getting rid of skipper Adam Gilchrist (27) to reduce the visitors to 67/2 in 9.3 overs.
Gilchrist, tried to recover from his lost form, but failed to make it big and got caught by Morgan while trying to hook Bhatia over mid-wicket.
Jacques Kallis gave Kolkata first breakthrough by scalping Punjab opener Mandeep Singh (25) after 45-run opening wicket stand.
Mandeep was caught Eoin Morgan after his 20-ball knock, studded with five boundaries.
Kings XI Punjab won the toss and elected to bat against Kolkata Knight Riders in the IPL clash at the Eden Gardens stadium on Friday.
Kolkata have brought in a leg-spinner in place of Iqbal Abdulla. Azhar Mahmood staged a come back in Punjab squad in place of Pomersbach, while Gurkeerat Singh replaced Parvinder Awana.
With just four points in their kitty from seven matches, KKR need a morale-boosting victory against Kings XI Punjab to remain in the mix.
If KXIP find themselves in fifth spot on the points table, two rungs above KKR, it is because they have punched above their weight to snatch a couple of improbable wins.
The biggest advantage of KXIP is that, unlike KKR, their relatively "unknown" domestic cricketers have chipped in with useful contributions to pull off victories against higher-rated opponents.
Kolkata Knight Riders: Yusuf Pathan, Gautam Gambhir(C), Jacques Kallis, MS Bisla, Eoin Morgan, DB Das, Rajat Bhatia, SMSM Senanayake, Sunil Narine, S Ladda, L Balaji
Kings XI Punjab: Adam Gilchrist(C), Mandeep Singh, Manan Vohra, David Hussey, Azhar Mahmood, David Miller, Gurkeerat Singh, Piyush Chawla, Praveen Kumar, Harmeet Singh, BA Bhatt