"We hope that the new pope will be more tolerant and far-seeing than the last one," said Muslim Brotherhood spokesman Mahmoud Ghozlan.
Argentine cardinal Jorge Maria Bergoglio, who has taken the name Francis I, is said to have been the runner-up in the conclave that elected Cardinal Joseph Ratzinger as Pope Benedict XVI in 2005.
"The previous pope had an approach (to inter-faith relations) that was at times provocative and his words appeared to attack the Prophet Mohammad," said Ghozlan.
A speech delivered by Benedict in Regensberg, Germany in 2006 sparked a deadly wave of riots in the Muslim world when he cited a 14th-Century Byzantine emperor who criticised Islam as a violent faith and labelled Mohammad "evil and inhuman", he said.
"We hope that Pope Francis is an open-minded and tolerant man, who will work for world peace and respect between religions, in accordance with Christ's teachings," Ghozlan added.
Ghozlan also said he hoped to see improved relations between the Vatican and Cairo's al-Azhar Institution, one of the highest authorities in Sunni Islamic learning.
Al-Azhar suspended a twice-yearly dialogue with the Vatican in January 2011 after Benedict XVI called for greater protection of Christians following a series of attacks on Christians and churches in the region.