Berbatov flicked the ball up and crossed in for Aaron Lennon, who saw his shot deflect off Jonathan Grounds and past his own keeper.
But Middlesbrough refused to lie down and Downing ensured they came away with a point after his 25-yard shot in the second half was deflected off Jermaine Jenas and flew into the corner of the net.
Middlesbrough could have broken the deadlock in the opening 60 seconds when Gary O'Neil crossed into Jeremie Aliadiere, whose shot was beaten out by Radek Cerny.
Afonso Alves knocked the rebound home but the effort was ruled out for offside.
Despite both teams having nothing to play for in the league, the game got off to a lively start and Tottenham also came close just 60 seconds later when Berbatov played the ball back to Jenas, who fired the ball just over the bar.
Robbie Keane was celebrating his 250th appearance and was involved in the heart of the action on 11 minutes when he crossed into his strike partner Berbatov, who headed the ball straight at Mark Schwarzer.
Jenas then made a superb run into the Boro half on 18 minutes beating three players, but saw his shot well kept out by Schwarzer.
The home side were finally rewarded for their early dominance when they opened the scoring on 27 minutes.
Berbatov produced a great piece of skill to flick the ball up and crossed into Lennon, who struck the ball off Grounds and past his own keeper.
Middlesbrough had a golden chance to grab an equaliser on 50 minutes when David Wheater rose the highest in the box but saw his header cleared off the line by Lennon.
Berbatov came close for Tottenham on 59 minutes when he saw his shot deflect off Emanuel Pogatetz and hit the post.
Tottenham were made to pay for their missed opportunities when Middlesbrough grabbed an equaliser on 69 minutes.
Downing's shot deflected off Jenas and flew past the stranded Cerny into the corner of the net.
Alves could have snatched all three points for Middlesbrough on 79 minutes following a mistake by Michael Dawson.
The Brazilian raced past the Tottenham defender but, with just Cerny to beat, he struck his shot inches past the post.