Left-arm spinner Sadia Iqbal claimed three wickets with Nashra Sandhu, Omaima Sohail and Fatima Sana all taking two each for Pakistan as Asia Cup champions Sri Lanka fell comfortably short of their target of 117.
Pakistan needed a crucial lower-order knock from their captain Sana to post a competitive total, having slumped to 84-8 after choosing to bat.
"We won the toss and we wanted a lot of runs but we didn't get. But we still achieved the target in the game," said Sana.
Sugandika Kumari had both Pakistan openers caught behind off her left-arm spin and the wickets continued to tumble at regular intervals.
Sohail made 18 and Nida Dar added 23 but Pakistan looked in danger of failing to reach 100 as Sri Lanka skipper Chamari Athapaththu picked up three wickets with her off-spin.
Athapaththu finished with 3-18 and Kumari took 3-19 but Sana's 30 off 20 balls, featuring three fours and a six, frustrated Sri Lanka and proved a match-winning contribution.
She was eventually dismissed by left-arm seamer Udeshika Prabodhani, who ended with 3-20.
Pakistan lost opening bowler Diana Baig to a calf strain after just one ball of the Sri Lanka innings, but her replacement Sana picked up the vital wicket of Athapaththu -- caught at extra cover for six.
"We all know Chamari is a good player, we needed that wicket to have a good start and we got it," said Sana.
Sohail bowled Harshitha Samarawickrama and Hasini Perera cheaply, but while Vishmi Gunaratne remained at the crease there was hope for Sri Lanka.
Kavisha Dilhari holed out to long-on off Sandhu while trying to increase the run rate to leave Sri Lanka reeling at 47-4 in the 11th over.
Nilakshika Silva earned a reprieve when she was trapped lbw from a delivery called a dead ball after Sandhu's towel fell out of her pocket.
But Gunaratne perished for 20 three balls later as she was caught just inside the boundary attempting to hit Sandhu straight down the ground.
Silva departed for 22 in the penultimate over after a top edge to short fine leg.
"We struggled to chase. We have to adjust to conditions," said Athapaththu.
"I hope we can bounce back. We bowled well. We have to bat really good too. We can't say wicket is slow and outfield is like this. We have to play positive cricket."
Neither team has ever made it out of the group stage in eight previous attempts at the T20 World Cup.
Pakistan face arch-rivals India in their next game on Sunday, while Sri Lanka take on tournament favourites Australia who are bidding for a seventh title in nine editions.
