The wicketkeeper-batter raised his first T20 hundred off 45 balls and remained unbeaten on 107 as a new-look Pakistan cruised to 3-197 for a seven-wicket win in the third Twenty20 with 16 balls to spare.
Earlier on Sunday, Bangladesh made 6-196, their highest total in a T20 against Pakistan, after the hosts' captain Salman Ali Agha won his third successive toss of the series but decided to chase for the first time.
Haris continued to follow Pakistan's new aggressive batting approach in coach Mike Hesson's first series as white-ball head coach. Haris hit seven sixes and eight boundaries and also became the first non-opening Pakistan batter to score a T20 century.
Pakistan had dropped seasoned pros Babar Azam, wicketkeeper Mohammad Rizwan and Shaheen Shah Afridi for the series to test their fringe T20 players as a build-up for next year's T20 World Cup.
Opener Saim Ayub, who struggled in Pakistan's first two wins with scores of 0 and 4, returned to form with 45 off 29 balls and put on 92 runs with Haris after Sahibzada Farhan (one) fell in the first over.
Power-hitter Hasan Nawaz, who scored Pakistan's quickest-ever century in a T20 against New Zealand in March off 44 balls, scored 20 of his 26 runs in fours and sixes before he gave spinner Mehidy Hasan Miraz (2-26) his second wicket.
Bangladesh couldn't capitalize on a 110-run opening stand between Parvez Hossain Emon (66) and Tanzid Hasan (42).
Legspinner Shadab Khan dismissed Emon in the 12th over and squeezed Bangladesh in the middle overs before fast bowler Abbas Afridi (2-26) choked the batting further in the death overs.
Afridi gave away only eight runs in his last two overs and picked up the wickets of Shamim Hossain (eight) and Towhid Hridoy (25) as Bangladesh could score only 46 runs in the last five overs and lost four wickets.
Pakistan won the first match by 37 runs and the second by 57 after Bangladesh failed to chase down identical targets of 202 runs in both games.
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