Smith smashed 101 off just 54 deliveries as Pune posted a challenging 195 for 3 after being put into bat but the Lions capitalised on the 93-run opening stand between Dwayne Smith (63 off 37 balls) and McCullum (43 off 22 balls) to pull off a narrow win at the MCA Stadium.
Captain Suresh Raina (34 off 28 balls) and Dinesh Karthik (33 off 20 balls) also made useful contributions as they stitched 51 runs together for the third wicket to take the Lions closer to victory.
The Lions needed nine runs from the final over bowled by Thisara Perera but they lost Raina and Kishan in consecutive deliveries, leaving them to get three from two overs.
The penultimate delivery fetched the Lions two runs and Faulkner took a single off the last ball to take the Lions to 196 for 7 and post their sixth win in seven matches.
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The Lions began their run chase in a flier with Smith and McCullum butchering the Pune bowlers to submission. The hard-hitting opening duo blazed to 72 without loss in the first six powreplay overs, hitting boundaries with impunity.
The opening pair took 24 runs off the third over bowled by Albie Morkel with McCullum hitting three fours and a six, followed by 15 and 14 runs from the next two overs.
Smith continued his aggressive batting and sent the next ball after McCullum's dismissal to the boundary to bring up his fifty off just 28 deliveries.
The Lions reached the 100-run mark in the ninth over but Smith was out in the next over as the West Indian inside-edged a Perera yorker length delivery onto the stumps.