Roy Hodgsons team almost took the lead after only five minutes when Luka Milivojevics shot from the edge of the penalty area hit the outside of the post with Cherries goalkeeper Mark Travers beaten.
At the other end Josh King lashed a shot over the bar before the Eagles took the lead in the 24th minute when Batshuayi sent a volley into the back of the net after a cross had deflected off Adam Smith.
The Belgium international scored again nine minutes later when he turned Aaron Wan-Bissaskas low cross home and, as Bournemouth defended carelessly, Palace added a third goal before half time when Travers parried Wilfried Zahas shot only to see the ball hit Jack Simpson and dribble into the net.
Eddie Howes team showed a great deal of character not to capitulate though and they scored a spectacular goal in first-half stoppage time, Jefferson Lerma smashing a 30-yard shot past Vicente Guaita in the Palace goal.
Palace started the second half brightly and Travers did well to save Andros Townsends curling shot and then James McArthurs effort.
However, the visitors scored again just before the hour mark, Jordon Ibe drilling a shot from the edge of the penalty area past Guaita.
Patrick van Aanholt made it 4-2 in the 65th minute when the former Sunderland defender played a one-two with Zaha before firing past Travers but Bournemouth hit back again eight minutes later - King netting from a corner at the back post.
However, Palace finally wrapped up the three points with 10 minutes remaining when Townsend turned the ball home after all the hard work had been done by Zaha.