City beat Bayern Munich 2-0 on Wednesday but could only finish third in Champions League Group A, due to Napoli's 2-0 win at Villarreal.
United, who had reached three of the last four Champions League finals, were also eliminated after a 2-1 loss in Basel.
City currently have a five-point lead over United at the top of the Premier League ahead of Monday's trip to Chelsea.
But Mancini maintained the domestic title race will be unaffected by the two Manchester clubs having to play in the Europa League, although the scheduling of the second-tier tournament could have an impact on domestic matters.
"I think that for the Premier League it changes nothing because both teams are in the Europa League," said Mancini.
"The difference is that the Europa League is harder because we play Thursday and we won't have a lot of time to recover for the games but this will be the only difference," the Italian added.
Mancini has always put City's problems in their first Champions League campaign down to a lack of experience.
But despite having a League Cup semi-final, as well as the FA Cup and the Premier League to play for, he maintains City will give everything in the Europa League.
"I think that for Manchester City its an important trophy because we need to win more trophies if it's possible."
"At the moment I can understand that the Europa League is not as important as the Champions League but for us it will be important and if we can go to the final it would be good for Manchester City.
"It's my opinion that we need to improve as a squad to play better in the Champions League.
"It was our first time in the Champions League, we got the worst group and it was difficult and with 10 points it's very strange that we didn't go through but that's it."
City won their final Group A match thanks to goals from David Silva and Yaya Toure either side of half-time.
But Napoli finally made their breakthrough in the second half at Villarreal to ensure City had to settle for third place.
Bayern coach Jupp Heynckes, whose side were already through to the knockout phase, made seven changes to the team that beat Werder Bremen 4-1 at the weekend.
He insisted that was due to a flu virus and is not expecting many of those players to return for the game against Stuttgart at the weekend.
"Due to the illness of a number of players I was forced to make changes," Heynckes explained.
"Toni Kroos and Arjen Robben are on antibiotics so we are not sure.
"Daniel van Buyten has gastric flu and with Thomas Muller and Mario Gomez there are signs of flu."
Heynckes, who said his side will not fear any of the other 15 teams left in the Champions League, added City had paid the price for a lack of experience.
"We were in a very difficult group and we?ve won it. That means we'll play away first and my team has performed excellently in the group stage.
"With the team and squad I have, I am confident that we can beat anyone.
"It takes a team years to develop at this level. City had a lot of new players in a short space of time but Roberto Mancini is slowly getting it together in terms of forming a good team.
"But in football today it takes time to develop. Good progress is being made but it will take experience and time."