Jefferson Lerma and Dominic Solanke were also on target as the Cherries picked up their second Sky Bet Championship victory in a row.
Duncan Watmore had briefly got Boro back on level terms with a well-taken solo strike to cancel out Billing’s first-half opener but Neil Warnock’s side are now six points adrift of the top six.
Bournemouth boss Woodgate, facing Boro for the first time since being sacked in June, was boosted by the return of Adam Smith and Junior Stanislas from injury and Lerma after suspension.
Colombian Lerma had the first sight of goal in the sixth minute when he scuffed a shot wide from 20 yards following a quick counter attack from the home team.
The Cherries took the lead eight minutes later, when Arnaut Danjuma acrobatically hooked the ball forward for Billing to send an unstoppable left-footed volley crashing into the net.
Boro were back on level terms in the 63rd minute courtesy of Watmore’s seventh goal of the season.
Yannick Bolasie played the ball into Watmore, who took a lovely deft touch to get it out from under his feet and caressed his shot beyond Asmir Begovic from an acute angle.
Boro’s joy lasted just four minutes before Bournemouth retook the lead from Lerma’s deflected shot from just inside the 18-yard box after Billing had crossed from the left.
Solanke sealed the points seven minutes from time, hammering in the rebound after Marcus Bettinelli had found his initial effort too hot to handle.
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