So much for the theory that the Britannia Stadium is a difficult place to come. On a cold and wet afternoon in the Potteries, and at the end of a helter-skelter game, Brendan Rodgers and Liverpool celebrated their first Premier League victory here and a hugely significant three points that lifts them back into the top four. Stoke City, however, could not have been more obliging opponents.

As ruthless as Liverpool were with their finishing, in particular the irrepressible Luis Suárez, who scored twice to take his tally for the season to 22, Stoke were guilty of some calamitous defending and contributed largely to their own downfall. Marc Wilson and Ryan Shawcross, Stoke's central defenders, will have nightmares about this game.

Shawcross may have been unfortunate with the own goal that gave Liverpool an early lead but it was impossible to have any sympathy with the manner in which the home team conceded a second, when the Stoke captain and Wilson got in a terrible mess.

Stoke, to their credit, hauled themselves back into the match with two goals in the space of six minutes at the end of the first half, when Peter Crouch and Charlie Adam each scored against their former club.

At that point the momentum was with Stoke but the complexion of the game changed again when Anthony Taylor, the referee, adjudged that Wilson had brought Raheem Sterling down.

There was definitely contact and arguably no need for Wilson to attempt to make the challenge but, at the same time, Sterling seemed to go to ground fairly easily. Rodgers described it as a "Spanish penalty" and admitted that Mark Hughes, the Stoke manager, was entitled to be aggrieved. When the dust settled, Steven Gerrard converted from the spot and Stoke were once again chasing the game – an unenviable position to be in against a team that counterattack as well as Liverpool.

The visitors' threat was augmented by the addition of Daniel Sturridge and the England international, who was making his first appearance in seven weeks after being sidelined with an ankle problem, had an instant impact when he set up Suárez's second. Although Stoke pulled one back through Jonathan Walters, whose weak shot should never have beaten Simon Mignolet, Sturridge restored Liverpool's two-goal lead two minutes later. Incredibly, there was still time for Crouch to hit the upright and Gerrard to be spared an own goal by a superb Mignolet save.

It was that sort of day. A pulsating game was full of goals and mistakes at both ends of the pitch. Rodgers was full of praise for his players, including Gerrard, who pulled the strings in a deeper role just in front of the back four, but if there is a concern for the Liverpool manager it is that his team looked vulnerable defensively. This was the seventh time in eight away games that they have conceded two goals or more.

Liverpool could hardly have wished for a better start. With only five minutes gone Aly Cissokho hit a left-footed shot from 25 yards that was veering closer to the corner flag than the goal until it took a huge deflection off Shawcross, to leave Jack Butland completely stranded.

Things went from bad to worse for Stoke when Martin Skrtel thumped a long ball from back to front that Wilson tried to head back to Butland. Shawcross, seeking to recover the situation, succeeded only in toe-poking the ball into the path of Suárez, who had the simple task of sliding it home.

Stoke had forced plenty of corners – six inside the opening 26 minutes – without ever really looking too threatening but they pulled a goal back when Peter Crouch got away from Kolo Touré and confidently headed home Marko Arnautovic's centre. Seemingly coasting, Liverpool began to look anxious and Stoke capitalised when Gerrard lost his footing deep inside his own half. Adam seized possession and took a couple of touches before cracking a sublime 25-yard drive that arrowed into Mignolet's right corner.

Then came the key moment in the game when Wilson, having squandered possession to Sterling, tried to atone but succeeded only in tangling with the winger and giving away a penalty. Gerrard sent Butland – who picked a bad day to make his first Premier League start – the wrong way from the spot.

Liverpool's fourth arrived when Sturridge carried the ball from inside his own half before choosing just the right moment to flick a neat pass into the path of Suárez, who curled a right-footed shot into the bottom corner. Walters, set up by Arnautovic, then saw his shot slip through Mignolet's arms. He quickly retrieved the ball but it was Liverpool who struck again. This time Suárez turned creator, his cross picking out Sturridge, who scored at the far post at the second attempt.

Man of the match Luis Suárez (Liverpool)