The Merseyside club looked to be coasting to their first Premier League win at the Britannia Stadium when Aly Cissokho's speculative strike went in off Ryan Shawcross and Suarez scored his 21st league goal of the season.
Stoke had other ideas, though, as former Liverpool duo Peter Crouch and Charlie Adam struck in a pulsating first half to peg Brendan Rodgers' side back.
Steven Gerrard put Liverpool back in front from the penalty spot on his 650th appearance for the club after referee Anthony Taylor controversially awarded a spot-kick for what appeared to be an innocuous challenge from Marc Wilson on Raheem Sterling.
Fit-again striker Sturridge was the architect for the prolific Suarez's second goal before Jon Walters set up a tense finale when he pulled one back.
But Sturridge then scored a goal of his own to seal the points and ensure Liverpool held on to leapfrog neighbours Everton into fourth place in the Premier League.
Defeat for Stoke was only their second at home in the league this season and Mark Hughes' side remain 12th, just four points above the relegation zone.
Sturridge was named among the substitutes after recovering from an ankle injury, while Kolo Toure paired Martin Skrtel at the heart of the Liverpool defence in the absence of Daniel Agger (calf).
Walters and Crouch returned to Stoke's starting line-up in place of Kenwyne Jones and Oussama Assaidi, who was ineligible to face his parent club. Goalkeeper Jack Butland made his first Premier League start as Thomas Sorensen was left on the bench.
Liverpool took an early lead at the Britannia Stadium last season and were in front after only four minutes on this occasion courtesy of a major stroke of luck.
Cissokho's long-range strike was never going to trouble Butland until it struck the unfortunate Shawcross and flew past the wrong-footed Stoke goalkeeper and into the net.
Stoke could do nothing about the opening goal, but they only had themselves to blame for Liverpool's second after 31 minutes.
Wilson was the chief culprit as he failed to get enough on his attempted header back to Butland and the alert Suarez slotted home after Shawcross failed to clear.
The home side responded and Crouch gave them a lifeline eight minutes later with Stoke's first headed goal of the season after he was picked out by Marko Arnautovic.
Butland got down well to deny Philippe Coutinho and Shawcross nipped in to clear the rebound, then Stoke were level just before the break when Adam let fly with a fine left-foot strike from 20 yards out that flew into the far corner.
Wilson had another moment to forget four minutes after the break when he fired the ball straight at Sterling and was then adjudged to have brought the winger down by referee Taylor when there appeared to be minimal contact.
Gerrard sent Butland the wrong way from the spot to restore Liverpool's lead on his landmark appearance for the Merseyside club.
Sturridge made his eagerly-awaited comeback 65 minutes in and the England striker took only five minutes to demonstrate his telepathic understanding with Suarez as his delightful backheel presented the Uruguay international with the opportunity to find the far corner with a clinical finish.
Walters was denied by a fine save from Simon Mignolet, but the Stoke forward beat the Belgium international with a tame finish six minutes from time.
Sturridge had the final say, though, as he was first denied by Butland, but kept his composure to fire home the rebound with his left foot. There was still time for Crouch to be denied by the post in an incredible game.
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