McLeish's side could have been ahead within the opening two minutes as Rangers skipper Lorenzo Amoruso totally misjudged a long ball from Franck Sauzee.
The ball dropped over his head and David Zitelli, making his home debut, was on to it in a fl
ash. The French striker immediately fired off a rising shot which Brown managed to push over.
Zitelli then hooked a shot from a Russell Latapy pass straight into the young keeper's arms.
Then a neat ball from John O'Neil allowed Ulrik Laursen to charge into the Rangers penalty area, the big Dane's parting shot blocked by Sergio Porrini before Zitelli sent a shot flashing over.
Tony Vidmar picked up the game's first booking in the 12th minute for a late challenge on Zitelli before Sauzee carelessly gave the ball away to Peter Lovenkrands, who found Allan Johnston in space only for the former Hearts star to trundle a low shot in the direction of Nick Colgan, who easily dropped on it.
Rangers had loud claims for a penalty when Lovenkrands brushed against Paul Fenwick and went down.
Johnston again had an opportunity to open the scoring when Billy Dodds cut the ball back into his path but, once more, his shot was more like a pass-back.
Latapy lifted the home fans off their seats with a typical mazy run only to shoot wide before Easter Road erupted in the 22nd minute as Hibs deservedly went ahead.
O'Neil spotted Zitelli in space, placed a slide-rule pass into his path and the Frenchman did the rest. The angle was tight but he managed to drill a low shot across Brown and into the far corner – his first goal at Easter Road.
Hibs might have had a second goal in the 25th minute when a Stuart Lovell run left him with only Amoruso to beat. But the Australian internationalist was caught in two minds and the chance was gone, Zitelli booked as he tripped Porrini.
Clark made a terrible decision when he penalised a perfectly fair Sauzee tackle on Konterman, the French ace winning the ball cleanly only for the referee to insist he had gone in two-footed on the Dutchman.
It was prime territory for Jorg Albertz, just 25 yards out but the big German blazed his shot wildly over the bar.
Former Hibs star Kenny Miller got the expected reception when he appeared for Rangers at the start of the second half in place of Johnston, while Andrei Kanchleskis came on for Porrini.
For Hibs, Dirk Lehmann took over from goalscorer Zitelli.
Hibs might have a penalty in the 50th minute when Lovell found Lehmann with a superb long ball, the striker managed to get off a shot which looped over the bar, referee Kenny Clark failing to notice Vidmar enjoying a tight grip of his shirt.
Only the woodwork saved Rangers going two behind on the hour-mark, Lovell being beaten at the edge of the opposition's penalty area by Vidmar and the ball breaking to Gary Smith 30 yards out. The defender sent in a great dipping shot which clipped the top of Brown's bar.
There was a real let-off for Hibs in the 63rd minute when Albertz slipped the ball to Barry Ferguson, who dispatched it into the net only to see the offside flag raised.
Hibs held on for a terrific victory despite a late scare when an Albertz goal was also ruled out for offside.
Hibs: Colgan, Laursen, Sauzee, Fenwick, Gary Smith, Latapy, O'Neil, Jack, Lovell, Paatelainen, Zitelli (Lehmann 46). Subs not used: Franks, Murray, McManus, Andrews,
Rangers: Mark Brown, Amoruso, Konterman, Porrini (Kanchelskis 46), Vidmar, Ferguson, Albertz, Tugay, Johnston (Miller 46), Dodds, Lovenkrands. Subs not used: Graeme Smith, Reyna, Wilson.
The full article contains 677 words and appears in Edinburgh Evening News newspaper.