The home side started brightly and Sterling's vicious cross forced David Marshall into an early save before Suarez fired wide.
Suarez came close once more in the sixth minute as his effort was deflected wide of the far post.
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Suarez then saw another effort deflected wide, but it wasn't long before he put the Reds in front. A neat interchange on the edge of the box involving Joe Allen and Jordan Henderson ended with Suarez volleying the ball home from 18 yards with Marshall stranded.
Five minutes later, neat trickery from Coutinho took him past two defenders from a corner and the Brazilian saw his low shot from a tight angle rebound back off a post, and Jon Flanagan saw his strike cleared off the line by the Cardiff goalkeeper two minutes later.
Liverpool doubled their lead four minutes before half time with an incisive counter-attack. Henderson freed Suarez, who squared the ball to Sterling, who was left with a simple finish to put his side 2-0 up.
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In the final minute of the half, Suarez bent a 20-yard effort into the far corner to take his tally to two and extend Liverpool's lead to three with 45 minutes played.
Coutinho dragged a 25-yard effort wide of the post after the break before Jordan Mutch reduced the deficit.
The 22-year-old peeled away at the far post to meet Peter Whittingham's deep delivery and nod the ball beyond Mignolet.
The goal acted as a wake-up call for the visitors, who began to cause problems for the home side with their set piece deliveries, with Mutch nodding over the bar at the far post.
Sterling and then Suarez almost calmed the nerves of the crowd, but the former was denied by Marshall before the Uruguyan struck the near post from a narrow angle.
Allen nodded over the bar from a corner and Marshall got down well to deny Suarez a hat-trick, but the afternoon belonged to the effervescent striker.