Hart starred despite conceding a 90th-minute goal to Jerome Boateng as City made a losing start to their UEFA Champions League campaign on Wednesday.
The England international's position has been questioned, with manager Manuel Pellegrini signing Willy Caballero in July to put pressure on Hart.
Cousillas praised Hart for his performance in the Group E clash, labelling the 27-year-old "excellent".
"Joe Hart is a great goalkeeper at international level and has had a great career here at Manchester City," he said.
"Last year at the end of the season he performed well and he has carried that into this season.
"Today he was excellent, he didn't make mistakes and we are happy not only with him but also Willy Caballero who is also a great goalkeeper so we have that position well covered."
Cousillas said his team were "sad and frustrated" after conceding the late goal at the Allianz Arena.
But he believes they are good enough to progress from a group that includes Roma and CSKA Moscow.
"For sure it is not a good start to lose the first game of a group stage but we still have five games as I said before and it is up to us," Cousillas said.
"We are a strong team at home and we have a strong squad and I am convinced that despite Roma's quality and the fact it is a strong group, the tightest group, we can still qualify and get good results in the coming games."
