Yeah, I think that belief in the manager has to play a huge part. Under Silva, they would have collapsed after Delph's sending off.
At the same time, the Keane sub is the kind of move that could so easily backfire on a less lucky general (how many times did we lament Moyes's inviting of pressure with a negative change?). However, much like how we've talked about the players being more motivated to play for Ancelotti, us fans are also more inclined towards seeing this as a direct response to Watford pumping long balls at Deeney, rather than just a hopeful flooding of the box with bodies. Another example is how I happily presume that he'll never throw Niasse on while defending a lead again, because he'll have seen how his inability to control the ball resulted in Newcastle being able to move up the pitch without fear of being caught out. If Silva had done that, I'd have held it over his head for months.
That may read as me being cynical or even critical, but it's honestly not. You only have to look across the park to see how galvanising it can be to have a unifying presence in the dugout.