Reeza Hendricks made 74 at the top of the order while captain Faf du Plessis top-scored with 78 as the Proteas made 192-6 after being put into bat - a record total for an international T20 game at Newlands.
Usman Shinwari was the pick of the Pakistan attack with 3-31, including taking two wickets in the same over, but the tourists came up short in their reply, finishing on 186-9.
Miller was a key figure in helping restrict the chase; he was twice involved in run outs and held four catches, his efforts seeing him named man of the match despite making just 10 with the bat.
The home side triumphed despite the absence of Quinton de Kock, who did not feature in Cape Town due to a groin injury that will also keep him out of the next two fixtures.
Gihahn Cloete took over wicketkeeping duties and also opened the batting, though it was Hendricks who dominated the early overs.
The right-hander hit eight fours and two sixes before becoming Shinwari’s third victim, Shadab Khan hanging on to a tough catch at long on as he moved to his right.
The same bowler had already removed Du Plessis and Rassie van der Dussen in the space of four deliveries, the latter departing for a duck as his attempt to steer Shinwari to third man only resulted in a spot of catching practice for wicketkeeper Mohammad Rizwan, filling in behind the stumps for the suspended Sarfraz Ahmed.
Miller was one of five wickets to fall for 35 runs but he produced direct hits in the field to dismiss Babar Azam (38 from 27 deliveries) and Rizwan, who was caught short of his ground when attempting to pinch an overthrow.
The South African batsman also crucially held on to the opportunity to dismiss stand-in skipper Shoaib Malik for 49 midway through a frenetic last over, making sure Pakistan’s late charge for victory came up short.
Hussain Talat had earlier made 40 but the visitors must now win when the teams next meet in Johannesburg on Sunday to keep the series alive.
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