Leader of the Greek left-wing Syriza party Alexis Tsipras, which emerged with the largest share of Sunday's vote but fell just short of an overall parliamentary majority, said the result "meant an end to austerity, humiliation [and] leaving behind five years of humiliation and pain".
European parliament president Martin Schulz congratulated the Syriza leader on "a historic victory for his party".
Schulz emphasised that, "The position in the European parliament is clear. Tsipras is the democratically elected prime minister of a member state […] with a full right in that regard to be respected".
He acknowledged that Sunday's election signalled "a major change for Greece", but would also have an impact at European level.
José Manuel Fernandes of parliament's EPP group said he was "hoping the eurozone won't be affected".
The Portuguese MEP added that going into government with the Independent Greeks, a party from the opposite side of the political spectrum to Syriza, was "not a very good signal".
"Tsipras is the democratically elected prime minister of a member state […] with a full right in that regard to be respected" - Martin Schulz
Gianni Pittella, president of parliament's S&D group said, "The political message from the Greek elections is clear: more social justice and an end to the troika and austerity."
He said the election result demonstrated the desire of the Greek people to bring an end to austerity and "troika diktats".
The S&D group president added that, "The renegotiation of the Greek debt, and in particular the extension of the terms of its bailout, should no longer be considered a taboo."
Pittella also expressed disappointment regarding Tsipras' decision to enter into a coalition with a party of the centre-right and said a government of "progressive forces" would have been preferable.
Parliament's ECR group was less welcoming of Syriza's victory, with group president Syed Kamall saying Greece "will have to face up to […] reality" and cannot change its economic course without the consent of other eurozone members.
He said that while "room to manoeuvre in the eurozone is limited", he hoped agreement between all parties could be found.
However, Kamall queried whether "the voters of Germany, Finland, Netherlands and other eurozone countries [will] be prepared to fund Greece via fiscal transfers if the new government decides to splash the cash".
ALDE president Guy Verhofstadt said the fate of Greece is intertwined with that of the EU and outlined his hope that the new government would "actively pursue" any renegotiation of the terms of its agreements "through cooperation and negotiations".
He also expressed doubts about Syriza's choice of coalition partner and any deviation from the current programme of austerity.
Verhofstadt said, "We are concerned about the fierce anti-immigration stance of the Independent Greeks [and] the toning down of austerity does not mean Greece is not in urgent need of drastic reform."
The election result was hailed by MEPs from parliament's GUE/NGL group of which Syriza is a member.
In a statement, GUE/NGL said that Tsipras' calls for renegotiation of "the unbearable Greek debt" cannot be ignored, while parliament's Greens/EFA group agreed that Syriza's electoral success would pave the way for a "political sea change".
Greens/EFA group co-presidents Rebecca Harms and Philippe Lamberts said in a joint statement, "This vote signals that the majority of Greek voters want to move on from Greece's outdated, decrepit party political system and towards genuine political transition."
They added that the result showed "resentment with the one-sided policies, sold as 'reforms' [that] focused solely on budget cuts".
Raymond Finch of parliament's EFDD group said that the upcoming elections in Spain and Italy and the prospect of anti-austerity parties making gains signalled that "the euro project itself is now dead in the water".
The swearing in of the new Greek prime minister coincided with a meeting of the Eurogroup, the informal body of eurozone ministers. Its president Jeroen Dijsselbloem confirmed the Eurogroup "stood ready to work with the new Greek government".
Dijsselbloem said the Eurogroup will discuss Greece's economic adjustment programme at its next meeting in February 2015.
Both Dijsselbloem and European commission president Jean-Claude Juncker have signalled that there is little support for any write-off of Greece's €240bn debt, a position Tsipras himself has called "unrealistic".