David Silva required 12 minutes of the second half to haul City back into the affair, converting from close range, and the Spaniard was at the centre of an equaliser, forcing home at the near post via a Glen Johnson deflection.
But, against a mounting comeback by the Manchester club, Coutinho found a pivotal winner, lashing home with barely 10 minutes remaining of the encounter to settle a tie to remember and unleash emotional scenes for Kopites.
Brendan Rodgers had maintained faith in exactly the same starting XI that successfully downed West Ham United last weekend, although vice-captain Daniel Agger returned to the group and took his place among the substitutes.
Before the match kicked off, the entire stadium observed an impeccable minute's silence in memory of the 96 supporters who lost their lives at Hillsborough, while the Kop formed a special mosaic reading '96 - 25 years' in tribute.
When the action got underway, the home side used the remarkable atmosphere inside the ground to catapult themselves into an early lead; Suarez's strength retained possession, his vision dissected the City defence and Sterling showed composure beyond his years to take a touch and curl around Joe Hart.
On the quarter-hour, the goalscorer adopted creator mode, scampering rapidly along the right flank and thrashing a dipping cross into the penalty area which Daniel Sturridge flicked towards the corner, but could only divert wide of the left post.
Having spent much of the opening 20 minutes limping, powerful midfielder Yaya Toure succumbed to an injury and was replaced by Javi Garcia - who instantly collected a booking - as the visitors struggled to contain the sheer force of the Reds.
The Kop rose to their feet en masse on 26 minutes as a Coutinho corner from the right arrived on a platter for Steven Gerrard but the captain's strong header was central enough for Hart to punch away.
Such respite was short-lived for City, however; from the subsequent corner, Gerrard whipped the set-piece to the near post, where Skrtel was waiting to hammer a looping header beyond the reach of Hart for 2-0 and delirium around the stands.
With moments remaining in the half, Coutinho worked himself into a prime shooting position after typically relentless pressing by the Brazilian and Jordan Henderson had retrieved the ball, but the No.10 pulled his low effort off target.
Silva had a diving header deflected wide on 44 minutes and mayhem ensued from the centre into the Liverpool box, but first Sterling and then Johnson stood firm to clear off the line. Simon Mignolet then sprawled brilliantly low to his right to beat away a fierce half-volley by Fernandinho to preserve a two-goal half-time lead.