Unai Emery's side dominated the second period after Valencia captain Ricardo Costa was dismissed in the 50th minute for picking up two bookings. However, they were unable to beat the outstanding goalkeeper Diego Alves, who saved Rakitic's penalty.
Denis Cheryshev was then also given his marching orders in stoppage time for two bookings, the latter for diving.
The first half was evenly matched, neither side creating too many clear-cut chances.
Carlos Bacca controlled a cross from Rakitic inside his area but was unable to get his shot away, letting one of the few opportunities of the first half go to waste. At the other end, Paco Alcacer blazed over the bar from outside the area.
Just five minutes into the second half, Valencia were reduced to 10 men, Costa shown a second yellow card for blocking a shot with his hand.
The Sevilla players were furious that the referee deemed the incident occurred outside the area, adamant they should have been awarded a penalty and not the free-kick they were given instead.
But when Sevilla were eventually awarded a penalty, when Cheryshev was bodychecked by Juan Bernat just after entering the box, they failed to take advantage.
Alves guessed correctly and dived to his right, saving Rakitic's spot-kick with his legs.
And the Brazilian goalkeeper came to his side's aid again a few minutes later, saving a close-range header from Vicente Iborra.
Valencia did not manage a single shot on target all game and once Costa had been dismissed, they merely sought to hang on for a point, which they gained due to Sevilla's lack of confidence in front of goal.
The hosts' bad luck continued right until the end of the game. They were denied what looked to be a clear penalty in the 89th minute when Fabrizio Fazio was bundled to the floor by Philippe Senderos, and Bacca sent the loose ball so wide of goal that it went out for a throw-in.
And in injury-time Cheryshev also went down in the area after a challenge from Jeremy Mathieu, but instead of being awarded a spot-kick, was shown a second yellow card and dealt the same fate as Costa.