The England global was forced off the pitch in his side's 2-1 defeat at the hands of Huddersfield Town on Saturday.
United claimed a 2-0 victory at the Liberty Stadium thanks to a double from Jesse Lingard.
Mourinho said: "We were in the game from the first minutes, pressing and forcing things, the attitude was good".
Swansea on the other hand targeted United defender Victor Lindelof, who was arguably at fault for both of Huddersfield's goals at the weekend. Left with a one-on-one, the England global curled his shot past Nordfeldt and into the bottom corner. Manchester United began to provide somewhat of a threat on the counter but it was the home side who had the best chance just before half time when young striker McBurnie fired over the bar. United had to be satisfied with the two goals and a place in the next round of the competition.
A much changed Manchester United side traveled to Wales to face Swansea City at the Liberty stadium.
Palace, now rock bottom of the league with just three points and two goals from their opening nine games, went ahead on 21 minutes when Bakary Sako drove home from Ruben Loftus-Cheek's assist.
Tuanzebe strode out of defence purposefully and spread a pass to Matteo Darmian on the right flank.
Sergio Romero made late stops from substitutes Tammy Abraham and Leroy Fer, but United eased into the last eight after sending on Nemanja Matic and Romelu Lukaku from the bench. It was hard to tell in this game whether the home side were simply toothless in offence or whether Mourinho's side were well drilled at the back, but another clean sheet was well deserved for the Manchester club and they now progress to the quarterfinals.
The Red Devils are looking to continue their League Cup title defence but do so without a host of first-team players, including Michael Carrick, Marouane Fellaini, Marcos Rojo, Paul Pogba and Zlatan Ibrahimovic.
Tuanzebe was the left centre-back in a back three for United and looked more than comfortable stepping up against Premier League opposition.