That was the last of 12 shots to Everton's one in the second half as we dominated territory and possession but couldn't find a way through.
In the end, Nikica Jelavic's 22nd minute strike was enough for the Blues to secure maximum points - adding our scalp to recent home wins against Manchester City and Chelsea.
Harry Redknapp made three changes to the team that lost to United in our last Premier League outing.
Scott Parker, Gareth Bale and Jermain Defoe - all three played in the FA Cup win against Stevenage - came in for Jake Livermore, Aaron Lennon and Saha.
Clear-cut chances were few and far between in the first half.
Seamus Coleman dragged an early effort wide and our first opening came on 18 minutes when Bale's throw released Defoe, but Johnny Heitinga got back just in time.
Brad Friedel saved well with his feet to deny Marouane Fellaini from a tight angle but the Toffees took the lead a minute later. Leon Osman turned away from Younes Kaboul and cut the ball back for Jelavic to finish first-time from 14 yards.
Bale tried his luck twice from 25 yards, both efforts curling just wide and Tim Howard easily gathered Luka Modric's low drive.
At the other end, Friedel turned away Jelavic's free-kick diving to his left.
The second half was one-way traffic - from Defoe's snapshot that forced a reaction save from Howard in the 46th minute to Saha striking the post in the 93rd.
Defoe blasted over, then fired at Howard after fine play from Modric and Scott Parker opened up the home defence.
Saha entered the fray for Emmanuel Adebayor in the 54th minute and we continued to press forward.
Modric fired over from the edge of the box, Kaboul was wayward with a free-kick from similar distance and Howard reacted well when Defoe's 22-yarder took a slight deflection.
Possession statistics showed 69 per cent in our favour midway through the half and Saha was next in line to try his luck, the French international did well to skip away from Phil Neville and Osman, only for Sylvain Distin to block with his head, typical of Everton's stout defending.
Rafa van der Vaart came on for Sandro and Friedel palmed away Everton's sole effort of the second half from Osman in the 71st minute.
It looked for a second that we'd found the equaliser in the 76th minute but the linesman's flag was up as Defoe volleyed home from close range.
Still camped in Everton's half, Modric's shot was blocked by Fellaini as we moved into five minutes added time.
Then the late agony. Van der Vaart's shot deflected off Heitinga and looped onto the roof of the net.
Modric played the corner out to the edge of the area for Benoit Assou-Ekotto, whose attempted shot found Saha eight yards out.
Saha managed to get the ball under control and get off his shot under pressure from Tim Cahill, but could only watch in disbelief as the ball struck the post with Howard helpless.
Everton (4-4-1-1): Howard; Neville, Distin, Heitinga, Baines; Coleman (Jagielka, 82), Fellaini, Cahill, Osman, Drenthe (Rodwell, 65); Jelavic (Stracqualursi, 78). Unused subs: Mucha, Hibbert, Gueye, Anichebe.
Spurs (4-4-2): Friedel; Walker, Kaboul, King, Assou-Ekotto; Bale, Parker (Livermore, 86), Sandro (van der Vaart, 69), Modric; Adebayor (Saha, 54), Defoe. Unused subs: Cudicini, Nelsen, Rose, Kranjcar.
Goal: Everton - Jelavic (22).
Referee: Mr M Halsey.
The script was written for Saha to score against his former club and the stage was set with 93 minutes on the clock, but the striker's close-range effort cannoned back off the upright.