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Considering Sasuke's surprised response to Sakura pushing herself to use the dregs of his chakra to save him, not just anyone could have pushed themselves to their limits to make sure the portal stayed open long enough to ensure Sasuke's escape the way Sakura had. Sakura's ability to give Obito more chakra stemmed from her deep loyalty to Sasuke, who she still loved. Sasuke is a person more moved by actions than by words, and Sakura's action of saving him surprised him, causing him to pause even in the midst of battle and thoughtfully consider Sakura. Sasuke now sees that Sakura's determination and perseverance to save him isn't just about saving the world, but that there is a personal connection there: she is able to push herself because of her feelings for him. Sakura had given it her all to save Sasuke, and this effort is not lost on Sasuke.
Sasuke had not yet seen much of Sakura's strength or determination in a situation where she was called to push herself to her limits, and when he finally does, Sakura earns his respect.
Thus Sasuke shows her his respect and appreciation by stopping her fall. It's not like it would have severely injured her if she had keeled over, after all. Sasuke doesn't only stop her fall; he clamps his hand around her arm and holds her close to him. He stands there for long enough that it becomes apparent that it's not a coincidental happenstance but an intentional connection with Sasuke pressing Sakura close to his side. He could have just as easily set her down immediately if the only thing he cared about was preventing her fall, but instead he lets the moment soak in for a bit.
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Something passes between them as they gaze deeply into each other's eyes, which are shimmering (in anime, shimmering eyes always mean deep emotion). Judging from the way Sasuke's eyes glitter as he looks at Sakura - possibly this is the first time Sasuke feels a romantic attraction to Sakura. Kishimoto makes a point of emphasizing Sasuke and Sakura staring in each other's eyes (dedicating two extra panels to focus on their eyes just to emphasize the connection), and it's meant to emphasize their relationship. Sasuke's activated Sharingan is further meant to emphasize the emotional intensity of the moment, as the Sharingan is a dojutsu triggered by their user's great emotion and eyes that reflect the heart.
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While this moment between them isn't enough to cause Sasuke to change his mind about his plans, it marks a shift in his relationship with Sakura. Sasuke finally begins to understand the extent of Sakura's devotion and love for him in a way he hadn't before. Judging by the shimmering of his eyes and his thoughtful, studious look at he gazes at Sakura, he is touched and intrigued by Sakura's unconditional loyalty, even attracted to it. Loyalty is a trait that Sasuke values very highly in those who wants to call him friend/family, and Sakura has proved herself loyal to him. Aside from that, she has just shown Sasuke that she is a strong shinobi in her own right, despite his mistake in underestimating her usefulness earlier. By doing so, she has acquired a new respect from Sasuke.
I've seen fans claim that Sasuke is only staring at the person who rescued him, claiming that it is perfectly normal to stare at your rescuer after they've saved you from a different dimension. But if that's true, why does Sasuke only stare into Sakura's eyes? Why doesn't Sasuke stare at Obito too if it's merely the interest of the rescued staring at his rescuer? Sasuke knows Obito uses Kamui, not Sakura, so why the interest in Sakura when Sasuke knew it was Obito's jutsu that saved him? Also, why do Sasuke's eyes shimmer (anime's way of telling the audience that the person is experiencing a great emotion)? And why does he hold her like that, standing by her side and holding her close to his chest instead of catching her by the back, which is more what I'd expect from a rescued person in a platonic connection with his rescuer. Putting an arm around someone's shoulder doesn't seem like a platonic way to stop a person from falling over. It looks like the romantic positioning of lovers.
