George Boyd scored the only goal just after the hour at Turf Moor, which moved the hosts to within one point of fourth-bottom Sunderland in the table.
Defeat leaves champions City five points behind leaders Chelsea, who play one of their two games in hand at home to Southampton on Sunday.
Pellegrini said: "When you have mathematical chances you must continue fighting. The most important thing is trust in what we are doing, try to win the next game and we will see at the end of the season where we finish.
"We are in second in the table so that is not bad. There are 27 more points that we must fight for. We know that every point we drop it is more difficult to reach the top of the table."
Pellegrini claimed his side had not played badly against Burnley, but that they had not created enough chances.
"I don't think we played really bad. We created three or four chances. Burnley had, I think, one shot on target and they scored a goal," he added.
"That was a normal performance that we need to improve and we will try to do it."
Pellegrini refused to criticise his players for their effort when asked if he felt his side had played badly or lacked desire, continuing: "I am sure that they tried to give everything but maybe it was not a good day creatively."
The City boss also revealed that midfielder James Milner was not risked on Saturday because of a knee injury, with one eye on the midweek UEFA Champions League last-16 second leg against Barcelona.
"Milner had a kick to his knee so it was important for him to rest. I hope that he will be fit for Wednesday," Pellegrini said.
