In driving rain, Barnsley posed few problems for Jason Tindall's men and at no point looked likely to launch a comeback.
Heralding the start of a lively opening, the visitors threatened to score inside the opening two minutes.
Adam Smith set Surridge free on the right and Barnsley goalkeeper Jack Walton left his net unguarded as he raced out of his area to try and cut out the danger, but Surridge rolled the ball beyond the far post from a tight angle.
Solanke then threatened after breaking free inside the area, forcing Michael Sollbauer to get back and make a vital challenge.
Barnsley's first effort on goal came from Alex Mowatt whose 20-yard strike was kept out by visiting keeper Asmir Begovic.
Bournemouth only had to wait until the 12th minute to get their noses in front, though, when Smith and David Brooks combined to set up Billing who beat Walton with a low drive from 20 yards.
Herbie Kane put a free-kick over as the hosts tried to reply but Bournemouth increased their lead in stoppage time at the end of the half.
Steve Cook won possession, allowing Solanke to make a driving run into the area before firing low across Walton and into the corner of the net.
Rico added a third in the 52nd minute with a superb 30-yard effort which flew into the roof of the net.
As Barnsley looked for a response, Callum Brittain fired in a shot from distance, but it was straight at Begovic who saved comfortably.
Barnsley head coach Valerien Ismael made a triple substitution shortly before the hour-mark, sending on Luke Thomas, Dominik Frieser and Clarke Oduor in place of Victor Adeboyejo, Conor Chaplin and Callum Styles.
However, the newcomers had little time to make an impact before Bournemouth made it 4-0 in the 68th minute.
Surridge seized on a loose ball and fired in a fierce shot which Walton got a hand to but could not keep out.
Cherries substitute Rodrigo Riquelme almost added a fifth goal soon after coming on as he tested Walton with a shot that the keeper parried at his near post.