The Turkish champions made an impressive start at the Rams Park Stadium, taking control of the contest with two early goals that left Stuttgart facing an uphill battle.
Stuttgart threatened first when Jamie Leweling struck an effort that hit both posts before returning to the field of play.
However, Galatasaray responded by taking the lead in the 12th minute. Following a pass from Mario Lemina, Roland Sallai delivered a cross from the right, and after the Stuttgart defence failed to clear the ball, Batrakov produced an excellent finish to make it 1-0.
The Russian forward struck again just three minutes later. This time, he ran on to Davinson Sánchez's pass behind the Stuttgart defence before calmly finishing past the goalkeeper to double Galatasaray's advantage.
Batrakov's quickfire brace made him the first Galatasaray player to score twice within the opening 15 minutes of a Champions League match, according to the club.
Galatasaray continued to threaten, with Victor Osimhen involved in promising attacking situations, while goalkeeper Uğurcan Çakır made important saves to preserve the home side's advantage.
Stuttgart attempted to fight their way back into the contest after the interval but struggled to break through Galatasaray's defensive organisation.
The visitors thought they had reduced the deficit in the 76th minute when Chris Führich found the net, but the goal was ruled out for offside.
Galatasaray held on comfortably to secure all three points and move up to six points in the Champions League league-phase standings.
Galatasaray strengthen Champions League campaign
The victory provides a significant boost to Galatasaray's European campaign after their earlier defeats to Sporting CP and Barcelona.
For Stuttgart, the defeat represents another setback as they continue their search for consistency in the competition.
Galatasaray will now turn their attention to their domestic campaign, with a league meeting against Amed S.F. next on their schedule.
Full-time result:
• Galatasaray: 2 (Aleksey Batrakov 12', 15')
• VfB Stuttgart: 0