Daniel Sturridge scored the first Premier League goal of the season, and Mignolet saved the first penalty, to give Liverpool a winning start to the season.
Before the game Luis Suarez was photographed smiling with his daughter, adding to the images from yesterday in training.
Jordan Henderson was preferred ahead of Joe Allen in central midfield. Daniel Sturridge was also named in the starting line up with Daniel Agger. Steven Gerrard led out three LFC debutants - Iago Aspas, Simon Mignolet and Kolo Toure.
Liverpool came out of the blocks quickest - spending the majority of the opening 5 minutes in Stoke's half.
With all 11 Stoke players behind the ball, the reds poked and prodded as they tried to grab the games first goal. Begovic, in the Stoke goal, was in inspired form, denying a host of chances from Aspas, Enrique and Coutinho.
A corner from Stoke however put the reds under pressure - Mignolet getting nowhere near Crouch's header - but Huth fortunately smashed his shot against the crossbar.
Daniel Sturridge put the ball in the back of the net from a fantastic Steven Gerrard free kick but it was ruled offside. Correctly.
Moments later and from a Gerrard corner, Toure rose highest to head off the crossbar, Sturridge couldn't control the rebounded shot which went high and wide.
Then Daniel Sturridge did break the deadlock. A left footed strike from outside the box was watched intently by the England striker, as it went right into the bottom corner.
1-0 to LFC and thoroughly deserved.
The reds continued their dominance in the second half - Coutinho enjoying his roving role and more so the movement of Sturridge and Aspas ahead of him - who were pulling Stoke's centre-halves all around the pitch.
Begovic too, sadly, was continuing his fine form to deny a host of Liverpool chances.
Aspas eventually made way for Raheem Sterling in the 72nd minute after a hard-working and impressive opening to his Premier League career.
As the game petered out Raheem Sterling gave away a free kick in our own half, Charlie Adam sent in the free kick and Daniel Agger handled the ball in the penalty area. With two minutes left Stoke had a chance to snatch two points away from Liverpool.
But Simon Mignolet stepped up with a fantastic double save - first correctly diving to his right to save the pen and then to react quickest for the rebound.
A great end for his debut, and Liverpool left Anfield with a fully deserved 3 points and a brief berth at the top of the league.
Anfield Online Man of the Match
Simon Mignolet (for the penalty save)