Set 238 to win, Pakistan reached their target with 23 balls to spare thanks to Sarfraz Ahmed's unbeaten 101 in Adelaide.
Ireland captain William Porterfield earlier made 107, but his wicket triggered a slide to 237 all out.
The Irish finish fifth in Pool B, behind West Indies on net run-rate, while Pakistan go on to meet Australia in Adelaide in the last eight on Friday.
Before the tournament, Cricket Ireland chief executive Warren Deutrom said it would be a "failure" if Porterfield's side did not reach the quarter-finals.
But for the third World Cup in succession, the Irish have acquitted themselves well, winning three games and beating two Test-playing sides in West Indies and Zimbabwe. England won only two matches, neither against Test nations.
Indeed, Ireland could have reasonably expected to go through with six points, only for their elimination to come as a result of several factors.
Heavy losses to South Africa and India damaged their net run rate, West Indies earned three big wins to boost theirs, while Pakistan recovered from two opening defeats to claim four successive victories.
This latest win continues the improvement of Misbah-ul-Haq's men, echoing the Pakistan side that began poorly when the tournament was last held in Australia and New Zealand in 1992, only to go and lift the trophy.
For Ireland, their creditable showing will be used as evidence by those who say the next World Cup should not be cut from its current 14-team format to only 10 sides.
The Irish still have the chance to qualify for that tournament as well as the opportunity to earn Test status through the 2017 Intercontinental Cup.
Victory over Pakistan at the Adelaide Oval would have seen them progress past the first round of the World Cup for the second time, but they were unable to take enough runs from a disciplined attack, then lacked the penetration and accuracy to make the chase uncomfortable.
Plenty of the Ireland top order got starts, but there was not enough support for Porterfield, who pulled, clipped and drove his way to the highest score made by an associate captain in a World Cup match.
However, the collective accuracy of the Pakistan pace attack was impressive and, when left-hander Porterfield miscued Sohail Khan to mid-on, the lower order was exposed.
Despite the sluggish nature of the wicket, the Irish regularly found trouble playing the short ball, with the last six wickets falling for 55 runs.
Ireland's collection of medium-pacers and spinners may have been able to use the conditions to their advantage, but regular width aided the Pakistan cause.
Sarfraz, dropped by wicketkeeper Gary Wilson standing up to Kevin O'Brien on 37, shared an opening stand of 120 with Ahmed Shahzad, whose cut shots benefitted from Irish inaccuracy.
When he skied Stuart Thompson to mid-on and Haris Sohail was run-out in a mix-up with Sarfraz, Irish hopes were revived.
But Misbah arrived to steady in a stand of 82, taking two sixes over the leg side before treading on his stumps off Alex Cusack with 30 still required.
That left the patient Sarfraz to complete his maiden one-day international hundred, with Umar Akmal belting Paul Stirling down the ground to bring Ireland's spirited campaign to a close.
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Ireland were knocked out of the World Cup by Pakistan, who qualified for the quarter-finals with a seven-wicket win.