Ugarte missed United's 3-0 defeat at Old Trafford a fortnight ago because his deadline-day transfer from Paris Saint-Germain didn't go through in time, and watched from the stands as Casemiro was substituted at half-time after his errors led to two goals.
The 32-year-old Brazilian was linked with a move to Galatasaray during the international break and spoke to Ten Hag who backed him to recover from the Liverpool setback.
Casemiro was substituted at half-time during Man United's 3-0 defeat by rivals Liverpool
But he warned that Casemiro must prove he deserves a starting spot now Ugarte's arrival has increased competition for places.
'Everyone can have a bad day at the office,' said Ten Hag. 'Of course I speak with him about the situation.
'It's not the first time I guess he's had to deal with a bad game, now he has to overcome this one as well.
'But it's normal in life you have highs and lows. In the life of a football player, there are many observing and have an opinion on this and you have to deal with it. He knows how to deal with it. Nobody's career goes only up.'
The United manager must decide whether to keep faith with Casemiro at Southampton on Saturday or give a debut to Ugarte, having admitted that his new signing was tired on Thursday when he arrived back from Uruguay's World Cup qualifiers against Venezuela and Paraguay.
Ten Hag confirmed that his conversation with Casemiro also included his role at the club, adding: 'I normally do talk to him and in this case I did as well. You definitely need him, for sure.
'But players in our club especially know that when you are playing at this level there is always competition.
'They also know you can only be successful when you have a squad that can cover. You need at least two players for every position. Sometimes three for two is okay when you have very robust players.
'He understands and all the players will understand. Then it's about what is the best fit for the team. They have to prove it – that's the key word – they have to take responsibility and win games for the club.'