The Bosnian, 28, came on as a half-time substitute in New York against a side made up of developmental players.
Mourinho said: “If we won 10-0 I would not be happy because it would mean it was too easy.
“Begovic was not sharp, a consequence of being tired from training. In the next game against PSG he will start.”
It was the first game of Premier League champions Chelsea’s three-match North American tour.
Begovic, an £8m signing from Stoke City this month, replaced Thibaut Courtois with Chelsea leading 1-0 courtesy of Loic Remy’s 12-yard strike.
Franklin Castellanos equalised after the interval before 16-year-old Tyler Adams put the hosts ahead in the 70th minute.
Sean Davis scored twice, either side of a low Eden Hazard drive, to give a Red Bulls side featuring only four players who have appeared in the MLS this season, victory in their first game against Chelsea.
“I was surprised by our fragilities,” said Mourinho. “I was surprised that we were not good enough in the second half to cope with that.”
Chelsea play French champions Paris St-Germain in North Carolina on Saturday and Spanish champions Barcelona in Washington DC on 29 July.
The Blues face Arsenal in the Community Shield on 2 August before beginning their Premier League campaign against Swansea on 8 August.
Chelsea first half: Courtois, Ivanovic (capt), Cahill, Zouma, Azpilicueta, Fabregas, Mikel, Traore, Oscar, Moses, Remy.
Chelsea second half: Begovic, Aina (Solanke 71), Zouma, Terry (capt) Azpilicueta, Ramires, Matic, Hazard, Loftus-Cheek, Oscar, Diego Costa.
Unused substitutes: Chalobah, Clarke-Salter.