The striker broke the deadlock after just 25 seconds when he was allowed to advance into the box before beating Robert Olejnik.
Nish doubled Hibs' lead in the 25th minute with an easy back-post header but Falkirk responded five minutes later through Michael Higdon, who nodded past Andy McNeil.
Paul Hanlon restored the home side's two-goal lead after the break, before Higdon notched his second of the game.
Hibs quashed any suggestions they were short of confidence in the first minute when they opened the scoring through when a Nish foray down the right flank culminated in John Rankin's deflected cross finding the forward on the edge of the box.
The towering target-man strolled past the Bairns rearguard unchallenged before coolly slotting the ball past Olejnik at the near post.
Hibs were clearly up for the game and continued to press the Falkirk defence in search of a second goal.
The visitors survived a scare in the eighth minute when captain Darren Barr was dispossessed.
Steven Fletcher emerged with the ball but dragged his shot wide from 25 yards when he had options to his right in Dean Shiels and Nish.
Falkirk's first chance came after 12 minutes when a loose ball fell to Gerard Aafjes but the defender swung wildly at the shot and missed the target.
Hibs should have doubled their lead when Fletcher teed up Nish inside the box but the striker missed the target from close range in the 17th minute.
Fletcher had a great chance moments later but his volley from 12 yards was superbly blocked by Olejnik.
It was no surprise when Hibs added their second in the 25th minute after dominating possession.
Paul Hanlon delivered a cross from the left flank and Nish was left with an easy header into the net at the back post after a slight flick-on from Fletcher.
Just five minutes later, however, Falkirk dragged themselves back into the contest.
Aafjes found Michael Higdon - who got in front of Rob Jones - with a great cross and the former Crewe player glanced a header past Andy McNeil.
The match was great entertainment value and Hibs should have extended their lead moments later.
But John Rankin somehow missed the target from 18 yards with just Olejnik to beat after a through ball by Alan O'Brien.
John Stewart then broke into the box for Falkirk but his low drive from an acute angle was well blocked by McNeil.
Hibs restored their two-goal advantage two minutes into the second half through Paul Hanlon.
Nish met Dean Shiels' free-kick at the back post and directed the ball back across the six-yard box for Hanlon to nod into the net.
Hibs keeper Andy McNeil was then called on to save Russell Latapy's low free-kick from just outside the area before the hosts came close to adding a fourth.
O'Brien found Fletcher with a neat through ball but the striker was denied by Olejnik.
Falkirk came back again and scored in the 53rd minute through Higdon.
Jackie McNamara's piercing pass found the striker on the edge of the box and he waltzed past his markers before hitting the ball high into the net.
The lively Fletcher then did well to create room for a shot outside the area but his left-footed strike crashed off the top of the crossbar.
Falkirk manager John Hughes introduced both Dayne Robertson and Mark Stewart for Russell Latapy and John Stewart in the hope the changes would lead to an equaliser for his side.
With time running out, it was the Bairns who upped the tempo as they put Hibs on the back foot and forced a succession of corners.
After a powerful run, Scott Arfield found team-mate Robertson in the middle of the park but the youngster's long-range strike missed the target.
Falkirk continued to pile on the pressure and only the woodwork saved Hibs with just a minute to go when substitute Stewart struck the post with a close range shot.