Toffees boss David Moyes opted to play with two in attack, with Yakubu partnered by recent signing Louis Saha.
The opening minutes saw plenty of attacking intent from both sides, without either goalkeeper being called in to serious action.
Saha sliced a shot wide when well placed on the left side of the penalty area and after 16 minutes Tony Hibbert's 25-yard drive forced a save from home keeper Rorys Aragon Espinoza.
Midway through the first half Standard took the lead after Tim Howard had parried a swerving Steve Defour strike from 25 yards.
The rebound fell to Milan Jovanovic, who lifted the ball across goal from the right for the arriving Alex Witsel to prod the ball home from a matter of a couple of yards out at the far post.
The rest of the first half saw the Toffees fail top break down a stubborn home defence, with Standard always ready to spring forward with their quick counter-attacking style.
Moyes opted to make no changes at half-time and in the opening stages of the second period the Toffees still looked short of ideas of how to get back into the tie.
Yakubu headed a yard wide of target just before the hour mark, but that was the closest that Everton came to a crucial breakthrough.
However, midway through the second half it was 1-1 on the night as Espinoza fumbled a corner from the left and as he tried to gather the loose ball, Phil Jagielka prodded it over the line from close range.
Everton's hopes of being able make the group stages lasted until 12 minutes from time, when Leighton Baines was adjudged to have brought down Jovanovic in the area.
The striker picked himself up to net from the penalty spot, with his effort beating Howard as he dived away to his right to make the score 2-1 on the night.
Late on Standard had chances to extend their advantage as the game opened up, but they had done enough to ensure more European misery for the Blue half of Merseyside.