The Red Devils confirmed the arrival of Eriksen on a free transfer on Friday before announcing on Sunday that they had come to an agreement with Ajax over a deal for Martinez.
The latter starred under Ten Hag in Amsterdam, while Eriksen also trained with the Dutch outfit earlier this year before he joined Brentford on a six-month contract.
Neither player will travel to Australia for the final two games of the club's pre-season tour, with their debuts instead potentially coming against Atletico Madrid or Rayo Vallecano towards the end of the month.
"The fact that we get him, we have him now in our squad, tells everything, because I think we were really selective. It was not the point to get players in, but the right players in. I'm really happy that we succeed with Christian Eriksen," Ten Hag told MUTV.
"The expectations are there. That's why we get him. I'm convinced he will fit into the squad and in certain facts of our game, he will help us. I'm looking forward to that.
"I think it's quite good. What do you say? He's an experienced player. He plays in countries abroad, in Italy, in Holland, a long time in England, so he knows the Premier League. It's an absolute advantage that we have him in.
"I think he is a magnificent football player and I think the fans will enjoy watching him, because he's creative, he has ideas. I think especially our strikers will also be happy because he's the one who can involve them in the game.
Ten Hag also branded Martinez "a warrior" when asked to describe the Argentina international, claiming that the defender will bring "aggressiveness" and "an attitude" to the team.
"He's a warrior and I think the fans will admire him. He has an attitude, fighting spirit. He brings aggressiveness in the game, in a good way. I think we need that. But also, he's also skilful and he can deal with the ball and he's left-footed," the Dutchman added.
"I think the Premier League is high intensity, lots of challenges. We need such players. I know from South Americans... I call it grinta. Controlled, aggressive. You need that in your game and with him we have such a player.
"Unfortunately we cannot have them here in the tour, because at the moment they arrive here we will fly back. So that doesn't make sense. We cannot wait to involve them in our squad and the fans also cannot wait.
"Every year it's the same as a trainer, you want your squad complete as soon as possible. But also fans look forward to new players and what is the squad and what can they contribute and they want an identification."
Man United have also signed Tyrell Malacia this summer, with the Netherlands international featuring in the team's two pre-season matches this summer against Liverpool and Melbourne Victory.
The 20-time English champions remain determined to sign Frenkie de Jong from Barcelona, but the deal is allegedly currently at a standstill, with the midfielder still keen to remain at Camp Nou.