Harry Redknapp's side were looking for their first league victory over their north London rivals since November 1999 and will not have a better chance after Eboue was shown a red card in the first half for two bookable offences.
Eboue can have no complaints over his dismissal after he kicked out at Luka Modric and had shown no common sense earlier in the game when he was booked for dissent.
But Arsenal showed all their fighting qualities to hold on for a point to leave Tottenham looking nervously over their shoulders at the bottom of the table.
As expected Robbie Keane made his debut for the home side as Tottenham's new captain, while Arsenal's new signing, Andrey Arshavin, had to settle for a place on the substitute's bench and Eduardo finally made a return to first-team action.
Arsenal could have broken the deadlock after just five minutes when Robin van Persie's ball picked out Eboue who burst clear into the box and fired a right-foot shot just past the post.
Eboue had the ball in the back of the Tottenham net on 13 minutes, but his effort was ruled out after referee Mike Dean adjudged the Arsenal midfielder to have fouled Jonathan Woodgate.
Modric produced a superb piece of skill on 27 minutes to pick out Aaron Lennon inside the box, but he dragged his shot wide of the goal.
Modric was involved in the heart of the action once again on the half-hour mark when he made a clever turn on the edge of the box and drilled a low shot which forced a fine save from Manuel Almunia diving low to his right.
Vedran Corluka almost gifted Arsenal the lead just 60 seconds later when he tried to let the ball run out of play. Samir Nasri refused to give up and stopped the ball crossing the line as he played the ball in to Gael Clichy.
The Arsenal defender fired the ball across goal but Carlo Cudicini just managed to stop Emmanuel Adebayor knocking the ball home.
Arsenal suffered a major blow on 34 minutes when Adebayor pulled up with a serious hamstring injury and had to be carried off on a stretcher.
Things got even worse for the away side just 60 seconds later when the game erupted into controversy when Eboue was shown a red card after kicking out at Modric.
Mike Dean had no hesitation in showing a second yellow card to the Arsenal midfielder.
Tottenham were determined to make the extra man count and came close again on the stroke of half-time when new signing Wilson Palacios fired in a blistering 25-yard shot which Almunia did well to tip over the bar.
Roman Pavlyuchenko then broke free inside the box on 49 minutes but wasted a golden opportunity by striking the ball high over the bar and his strike partner Keane went even closer seven minutes later when his header skimmed the top of the bar.
Tottenham were almost made to pay for all their missed opportunities when Alex Song came within a whisker of putting Arsenal in front.
Van Persie's cross picked out the Arsenal midfielder who was unmarked inside the box but he dragged his shot inches past the post.
Nicklas Bendtner could have grabbed a late winner for Arsenal in stoppage time but the substitute's shot was superbly tipped over the bar by Cudicini.
Modric had an even better chance seconds later when he was clean through on goal but his shot was well-saved by Almunia.