The 24-year-old put Real ahead in Wednesday's Group B encounter with a stunning free-kick that dipped neatly over the wall to fool goalkeeper Eray Iscan in the 37th minute.
Bale's superb strike, which marked his sixth goal since joining from Tottenham in September, also served as a much-needed boost following Sergio Ramos' red card 11 minutes earlier.
Umut Bulut pulled Galatasaray level a minute after Bale's opener before Real clinched a comfortable victory courtesy of second-half goals from Alvaro Arbeloa, Angel Di Maria and Isco.
The 2012-13 La Liga runners-up had already secured a last-16 spot in the competition prior to Wednesday's game and Bale feels that Carlo Ancelotti's men can go all the way and win the tournament.
"We showed character and companionship to play a man short," he told the club's official website.
"There are no limits to what we can achieve in the competition.
"We have many chances to win the Champions League this year.
"I am satisfied by the result and by the goal. It is one of the best goals of my career."
