Everton fan Jon Walters gave Stoke the lead on 44 minutes with an unmarked close-range header, before substitute Coutinho drove home from just inside the area with 20 minutes remaining.
And two minutes later Liverpool were ahead as Firmino’s superb half-volley flew into the roof of the net from the edge of the box as the visitors broke.
The result means Liverpool stay third, nine points ahead of fifth-place Arsenal, who have three games in hand. Stoke are 13th, eight points off the drop zone.
Jurgen Klopp was forced to start youngsters Ben Woodburn and Trent Alexander-Arnold at the Bet365 Stadium, with Coutinho and Firmino both on the bench following illness and a groin problem respectively.
It was a first half to forget for Klopp as Liverpool struggled to get into the game, though Stoke also struggled to create many clear-cut chances.
The first chance fell to Stoke after Dejan Lovren made a hash of a clearance on the stretch, but Marko Arnautovic could only volley into the side-netting at an angle 10 yards out.
Before Stoke took the lead with a minute of the first half remaining, Mike Dean waved play on as Woodburn, Liverpool’s third-youngest starter in Premier League history, went down under pressure from Erik Pieters, though a penalty award would have been soft.
Stoke went up the other end through Xherdan Shaqiri on the right, and his cross at the byline found the unmarked Walters to head past Simon Mignolet from six yards after Lovren and Joel Matip went missing. The goal was Walters’ seventh against Liverpool in the Premier League.
Both Woodburn and Alexander-Arnold were replaced at half-time by Brazilians Coutinho and Firmino, but Stoke should have been 2-0 up after Gini Wijnaldum’s misjudged header fell to Charlie Adam, whose close-range effort was superbly touched wide by Mignolet.
The chances then flooded Liverpool’s way around the hour mark; first, Firmino’s shot into the ground was tipped wide by Lee Grant, before Coutinho’s low, controlled effort from the edge forced Grant into another good save.
Lovren then got up highest from a Coutinho corner, but his header came off the top of the crossbar and clear.
Liverpool were finally level with 20 minutes remaining through Coutinho, whose controlled effort into the bottom left corner came after Glenn Whelan’s poor clearance inside the area.
Just two minutes later Liverpool had completed their comeback, as Wijnaldum’s lofted ball from deep found Firmino, who went early with the shot and half-volleyed superbly into the back of the net from 20 yards.
Stoke were nearly level seconds later through Saido Berahino, but Mignolet somehow kept out his effort with his thigh from point-blank range after Arnautovic’s teasing cross.
Liverpool fought to see the game out, handing them their first away win in 2017.
