I'll be the first to say that I am biased in this and as such we're not going to talk about who wins... at least not directly.
Now Tsunade is known to have monstrous levels of strength not just from Chakra Enhanced Strength but from natural strength as well. Which would be why Kabuto was so keen on cutting her muscles. Now some of you may say, it is ass backwards to imply Hashirama was talking about a younger Tsunade. But as far as I'm aware super strength is like a kkg either you have it or you don't. If Tsunade had not always had super strength how did she magically acquire it? Lifting and maneuvering Bunta's sword was not the peak of her physical strength as she managed to do so seemingly effortless. Keeping in mind that bijuu sized summon has himself lost grip of his weapon, this feat alone speaks volumes for Tsunade's base strength.
Nextly, some find it ridiculous that someone who isn't even a genin can possess such abilities. The first thing I would say in this regard is merely thinking about the bell test. Hiruzen was known as a top tier jounin, and unlike Kakashi's first bell test, the intent of Hiruzen's bell test was to actually get the bell. Thus we can presume that the battle between the soon to be sannin and hiruzen would be more like Naruto and Sakura's second bell test. I'm not saying that Tsunade and Orochimaru beat Hiruzen, but there skills were great enough to get bells from a ninja like hiruzen when they were still very young. Certainly younger than the time skip characters.
The next reason it isn't so difficult to believe that Tsunade had super strength as a youngster is because prior to Sakura's training with Tsunade, she was still academy level. The only reason Sakura made it out of the academy was her natural affinity for chakra control and her intelligence. In terms of skills and abilities at no point did she show any spectacular abilities that would be beyond the grasp of an academy student. Thus, Tsunade learning CES as an academy student would be a stretch. Especially considering her natural inclination for strength attacks, the next step for someone with super strength would be increasing that strength. And Tsunade seems to have always had a knack for chakra control as well. As Tsunade would have had no chakra being diverted from anywhere, since she was already young, and had no yin seal all of her large chakra reserves could go directly into her attacks back then.
Now the most obvious thing that is said is that Kishimoto meant for Hashirama to be referring to Tsunade's strength in her prime. But this idea is curious. Firstly that would be implying that Sakura's CES> Tsunade's Natural Strength + Tsunade's CES. Nextly, Kishi had not long before drawn Tsunade in Hashirama's mind as only a little girl. Indicating that Hashirama died before he would have seen Tsunade in her prime. Further, Kishi in no way tried to inform or update Hashirama's knowledge. He could of had it so that Hashirama had seen her manifest her strength in some way. Or even better of all the shinobi on the battlefield several of them had seen an adult Tsunade in action. Sakura, Shizune, Kakashi, Hiruzen, Minato and Chouza for example. Shizune got a nice panel next to Hashirama's text of exclamation and if she so desired she could have harped on how strong sakura was just like everyone else but this was of no interest to her. And Time-skip Sakura is completely terrified of Tsunade's strength despite her showing results similar to Part 1 Tsunade's strength which indicates there was something far more fearful than part 1 Tsunade's showing. That's war arc Tsunade's showings.
Now personally I don't actually know if I believe that it was suppose to be young Tsunade that Sakura was being compared to, but it doesn't create plot holes or logical inconsistencies. So people who absolutely do 'believe it' , aren't necessarily crazy or stupid. They just interpret the manga differently. And unfortunately, Tsunade didn't much in the way of fights in the manga.
Also in straight CQC, Tsunade has shown herself to be more capable feat-wise that Sakura, while drunk off her ass.