Aritz Aduriz and Inaki Williams done the damage for the Basque club, who had Raul Garcia and Ander Iturraspe sent off after Lionel Messi pulled a goal back.
The Colombian would have impressed his coach on his first anniversary at the Bernabeu as he netted a double to put Jorge Sampaoli's men to the sword.
The England international found the back of the net either side of the interval to help Spurs move up to third in the Premier League and seven points behind Antonio Conte's men.
The Cherries raced into a 3-0 lead but goals from Alexis Sanchez, Lucas Perez and Olivier Giroud completed a remarkable comeback for the Gunners on Tuesday.
The Frenchman scored an incredible effort to put the hosts into the lead and Alex Iwobi made the points safe against Sam Allardyce's toothless Eagles.
Mauricio Pochettino's men were in complete control for the majority of the fixture against the Hornets, with Dele Alli and Harry Kane both at the double at Vicarage Road.
Jurgen Klopp's men remain within six points of leaders Chelsea after seeing off the challenge of the Citizens at Anfield, with an early goal making the difference.
The world's most expensive footballer scored an excellent header a minute after Anthony Martial had equalised following Grant Leadbitter's opener at Old Trafford.