The world's eighth-ranked team plundered 162-6 in Geelong, blasting 102 of them off the final 10 overs, to set a big target which the Netherlands were unable to reach, finishing on 146-9.
But the men in orange still went through after United Arab Emirates won their first-ever game at a T20 World Cup to upset Namibia by seven runs.
The Africans would have joined Sri Lanka in the next round rather than the Dutch had they won, but against the odds, the UAE restricted them to 141-8 as they chased 149 for victory.
For the Netherlands, it is the second time they have got this far after also making the grade in 2014, when they topped their group and then sensationally beat England.
It is the third successive time Sri Lanka have reached the Super 12 and they will be hoping to repeat their heroics from 2014, when Dinesh Chandimal skippered them to the title.
But it was a rocky road to get there, stunned by Namibia in the tournament's opening game before crushing UAE then the Dutch.
"We knew that we could be the number one in this group, but we missed out in the first match," said Sri Lanka skipper Dasun Shanaka.
"The boys have done well though, especially the bowling group. I knew the wicket was two-paced, so we wanted to bowl well in the first 10 overs and then control the game."
After winning the toss and choosing to bat, Sri Lanka started slowly with Pathum Nissanka and Mendis plodding to just 22 off the first five overs.
Great moment
Nissanka was dropped in the fifth over, but came undone on 14 when hostile pace bowler Paul Van Meekeren returned to the attack and sent down a yorker.
Dhananjaya de Silva followed him back to the pavilion the next ball, out lbw, as Sri Lanka lived dangerously with countless chances falling just short of the fielders.
Charith Asalanka and Mendis knuckled down in a 60-run stand to step up the run rate before Asalanka was caught behind off all-rounder Bas de Leede for 31.
Mendis brought up his ninth T20 half-century with a six, then smacked successive sixes in the penultimate over, before he was out going for another boundary, almost carrying his bat.
Despite missing injured pace spearheads Dushmantha Chameera and Pramod Madushan, a Sri Lankan attack with replacements Lahiru Kumara and Binura Fernando kept the Dutch at bay early on.
Opener Vikramjit Singh was out for seven off Maheesh Theekshana, then Kumara got the big wicket of rising star de Leede for 14.
Opener Max O'Dowd (71 not out) gamely hung on but when captain Scott Edwards was bowled by Fernando after a quick-fire 21, they slumped from 100-4 to 109-8 and the game was all but over.
In the later match, Zahoor Khan and Basil Hameed grabbed two wickets each as the UAE reduced Namibia to 69-7 in the 13th over before late fireworks from David Weise (55 off 36 balls) set up a tense last-over finale.
While a crushing disappointment for Namibia, getting a first-ever win was a huge confidence booster for UAE.
"A great moment for UAE cricket. We had been doing well in patches, but today everything clicked together," said skipper Chundangapoyil Rizwan. "This win is really big for UAE cricket, it can tell the world that we are here."
The final two places in the Super 12 will be decided on Friday with Scotland, Zimbabwe, West Indies and Ireland all still in contention.
