Mitchell Santner led New Zealand's recovery with a captain's knock of 47 off 26 balls after coming in with his side in huge trouble at 84-5 to power them to a competitive 168-7 at the R Premadasa Stadium.
Sri Lanka managed 107-8 in reply, never recovering from a top-order collapse to succumb to their second defeat in the Super Eight stage of the 20-team tournament.
Rachin Ravindra (4-27) and Matt Henry (2-3) impressed with the ball for New Zealand.
"It's embarrassing to disappoint the home crowd, they've been very supportive," Sri Lanka captain Dasun Shanaka said.
"We wanted to restrict them to 130 but, to be honest, they played really well so we must give them credit."
Having lost to England in their first Super Eight match, it was a do-or-die game for Sri Lanka, and they initially showed a sense of urgency with the ball after electing to field.
New Zealand start well
New Zealand got off to a strong start before losing openers Finn Allen and Tim Seifert in successive overs.
They were 75-2 in the 10th over before the Sri Lankan spinners wreaked havoc to reduce them to 84-6.
Maheesh Theekshana (3-30) removed the well-set Ravindra (32) and Mark Chapman in the same over to turn the screw.
Santner joined Cole McConchie, who made 31 not out, in a rapid 47-ball stand of 84 that proved decisive.
Santner smacked four sixes as New Zealand plundered 70 runs from the last four overs to ensure their bowlers had a decent total to defend.
Henry bowled Pathum Nissanka for a duck with the first ball of the Sri Lankan innings and deceived Charith Asalanka with a slower delivery in his next over.
Player of the match Ravindra then whittled out Sri Lanka's middle order to derail the chase.
The left-arm spinner sent down a two-wicket over in which Kusal Mendis and Pavan Rathnayake were stumped to reduce Sri Lanka to 29-4 in the ninth over.
Ravindra also dismissed Shanaka for three to effectively seal their victory.
Kamindu Mendis (31) and Dunith Wellalage (29) offered brief resistance but Sri Lanka were never in the chase.
"It's one of those wickets where, if you can get in and give yourself a bit of time, you can cash in," Santner said.
"It was pretty tough to start out there, so to hit 160 was nice."
Group Two leaders England have qualified for the semi-finals with four points from two matches.
New Zealand, who came into the match with one point from their washed-out match against Pakistan, have three points.