Ghanaian striker Tagoe put Ettifaq ahead after six minutes and Bashir made it 2-0 in the 16th minute before Carlos Renato threw the visitors a lifeline by pulling a goal back in stoppage time at the end of the first half.
But the hosts pulled away after the break as Bashir headed in his second of the game midway through the second half before Tagoe rounded off the scoring nine minutes from time.
With their victory, the Saudi side moved into first place in the group on six points, two ahead of Iranian side Sepahan and Uzbek champions Bunyodkor. Al Shabab are last with three points.
Ettifaq quickly asserted their authority, taking an early lead when Tagoe outjumped the Al Shabab defence to head home a corner from the left by Sayyaf Al Bishi after six minutes.
The home side continued to dominate and extended their advantage in the 16th minute as Tagoe teed up a pass on the edge of the box for Bashir who lashed a right-footed shot into the bottom right corner of the net.
Al Shabab battled back after their disastrous start and threatened in the 26th minute with a curling Carlos Renato free-kick that Ettifaq goalkeeper Hussain Shaian turned aside for a corner.
Renato had another opportunity from an indirect free-kick in the Ettifaq box but sent the ball into the defensive wall after it played to him by Marcus Assuncao.
In stoppage time, Essa Al Muhairi’s shot slammed against the crossbar but in the ensuing move, Al Shabab finally found the net as Renato fired a powerful low shot with his right foot past Shihan.
Ettifaq reasserted themselves in the second half and went close to adding a third around the hour mark when Bashir’s free-kick shaved the left upright and Tagoe saw his effort stopped by goalkeeper Ahmed Mahmoud.
A third goal did arrive in the 67th minute as Saod Al Kaebari’s free-kick from the left was met by a leaping Bashir, who put a looping header past Mahmoud.
Al Shabab should have pulled a goal back in the 71st minute as Marcus Assuncao’s powerful free-kick rocked the crossbar.
But the home side put the game out of reach in the 81st minute when Bashir’s cross from the right was met by Tagoe, who put a first time shot past Mahmoud to round off the scoring.