Yes Sanji most likely defeated P1 reason being that Sanji's motivations for engaging Page One in the first place were first and foremost to prevent the later from destroying the town & hurting the townsfolk in a search for him and then to lesser degree reduce the strength of Kaidou's forces by taking out one of his stronger members. After the Sanji fight, there is no sign that the destruction of the capital continued, and we also haven't seen any trace of Page One since then. Not to mention Sanji bailing on that fight would have defeated the purpose of him engaging Page One in the first place. Moreover, we know that the search for Sanji initially started with 3 pursuers however, that number was reduced to the 2 in the following scenes after the fight implying Page One has already been taken out and is unable to continue the search for Sanji.
That said, seems like most people for the most part agree that Page One has been taken out, except that some folks believe that it was Drake that took Page One out and not Sanji coz he's a marine spy but why would Drake have suddenly done that? There was no incentive for Drake to have take Page One out at that point as he didn't even know the SHs and Law were on Wano and only found that out later. Moreover, why would Drake have born the risk of taking out Page One given Hawkins who isn't in cahoots with him was also in the vicinity. That notion is very weak in comparison to the notion of Sanji having taken out P1 as 1.) Sanji had strong motivations to have done so and 2.) Sanji was clearly dominating him, indicating the had the ability to carry out what he said he was going to do.
Lastly,
@kurwa has already kindly posted it but even Oda has said that the anime adaptation is different from the manga so there's no sense in using it to draw conclusions on the manga as the anime staff can take as many liberties with the anime as they want. I mean it's one thing for the anime to add fillers to fill in for gaps in the events but in the current one piece episode, they even went as far as to change the sequence of events in that fight to fit their narrative (i.e. I'm referring to the bit about Sanji saying he was going to have Usopp and Franky change the design of his raid suit).
In the Manga
Sanji takes P1's attack which destroys a large part of the town and then as result states that he has to end the fight quickly as to not further endager the town. Plummets up to the sky and then comments on having ussop and franky change the design of his raid suit, the scene the cuts of to Law and the others escaping the town and then we see Sanji land a devastating attack, which was felt all the way from outside the town buy the escaping trio of Law, Ussop and Franky, to which Page One with Page One screaming in pain.
In the Anime
Sanji takes P1's attack which destroys a large part of the town and then as result states that he has to end the fight quickly as to not further endanger the town. Plummets up to the sky, and then the scene the cuts of to Law and the others escaping the town and then we see Sanji land a devastating attack, which was felt all the way from outside the town buy the escaping trio of Law, Ussop, Franky, the fight then continues with Sanji and P1 and then after Sanji lands the O-soba mask whatever attack with Page One tanking it lolz....He says that P1 is very durable, and then mentions that he'd like Usopp and Franky to change the design of his raid suit before bailing.
It wasn't even like Sanji bailed because Drake and Hawkins came, he just decided to bailed just coz lolz which makes zero sense. I can only imagine that the anime wants to utilize Page One's character a bit more which is why they ended it the way they did given Oda didn't explicitly show Page One taken out. Maybe they thought it was a waste to have his character end as quickly as he was introduced and want to utilize him in some sort of filler scene or something?