Galatasaray must beat Juventus, which beat Copenhagen 3-1 on Wednesday, in a winner-takes-all game to advance.
Madrid featured a makeshift starting lineup with Ronaldo and Sami Khedira injured, and Luca Modric suspended.
Jose Rodriguez led the attack in front of Bale, Isco and Angel Di Maria, while Karim Benzema and Xabi Alonso were left on the bench the latter in place of Asier Illarramendi. Carlos Casimero was given his first Madrid start in midfield.
"With Cristiano our team is quite different," coach Carlo Ancelotti said. "The team reacted well. We played with courage and personality, we played really well. It's not always easy to play with 10."
Bale wasted no time in trying to lay down his claim as the catalyst up front, but he side-footed wide four minutes in with only goalkeeper Eray Iscan left to beat.
There was little excitement until Ramos was sent off for getting a hand to Bulut on the edge of the area as the last defender, and Selcuk Inan curled the ensuing free kick straight at goalkeeper Iker Casillas.
Bale gave Madrid the lead when he floated a free kick that flew high before dipping over Iscan. Only Ronaldo could probably emulate the Wales international's goal.
But the momentum didn't last, with midfielders Casemiro and Illarramendi failing to set the tempo, with Ronaldo watching on from the stands.
"The first half was not our best," Ancelotti said.
Galatasaray striker Didier Drogba perfectly toe-poked with the outside of his foot to meet the run into the box of Bulut, who slid his shot under the onrushing Casillas to equalize.
"The first half we were perfect. The second half, I don't know," Galatasaray coach Roberto Mancini said. "I need to talk to the players to understand what happens. We changed everything.
"This is strange because we had a big chance to win here."
Madrid was the better team after the break and Di Maria found Arbeloa in the area with a cross from the left, the Madrid defender dragging his foot back to meet the pass in the 51st minute and guide home his first goal in three years.
Alonso came on as Madrid steadied and, after a near-miss from Arbeloa, the home side scored a deserved third as Di Maria completed a clever buildup in the 63rd.
Modric's absence gave Isco a rare start, and the Spain international dribbled free inside the area before coolly sliding home in the 81st.
Apart from setting up the Galatasaray's opener, Drogba hardly made his presence felt until the 71st when Casillas leaped to tip his driving header over the bar.
"We knew before this game that the crucial game would be against Juventus in Istanbul," said Mancini, referring to the last round.
Juventus has six points while Galatasaray and Copenhagen both have four.
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