Maqsood's 90-ball knock took Pakistan home with 11 balls to spare but their top order looked far from convincing as they chased 247 runs for victory.
Having beaten the former champions in the 1999 World Cup, Bangladesh rekindled hopes of another upset by reducing Pakistan to 199 for six in the 43rd over but were thwarted by Maqsood who hit nine fours and two sixes.
Bangladesh posted a competitive 246 after opting to bat first but were all out with one delivery to go in their innings.
They lost two quick wickets but Tamim Iqbal (81) and Mahmudullah (83) added 168 runs to consolidate the innings.
Iqbal's 109-ball knock will please captain Mashrafe Mortaza the most as the left-handed opener was returning to action after undergoing knee surgery in December.
Down the order, all-rounder Shakib Al Hasan scored a brisk 31 before Mohammad Irfan (5-52) ripped through the middle and lower order.
Towering Irfan's five-wicket haul was a rare positive in Pakistan's otherwise wayward bowling effort, underlined by the 17 wides they conceded.
Pakistan's top order did little to convince their fans and the 1992 champions were 52 for three in the 16th over before Maqsood and Harris Sohail (39) steadied the innings with a 51-run stand.
Umar Akmal fell after contributing 39 but Maqsood guided them home.
(Reporting by Amlan Chakraborty in New Delhi; editing by Michael Hann)
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