With the PM away on his EU renegotiation tour, the shadow first secretary of state clashed with the chancellor for the first time, quoting Tory MPs unhappy at his progress.
Mr Osborne questioned the wisdom of quoting backbenchers in a dig at the unrest in Labour ranks.
He also pledged £50m in flood relief.
Mr Cameron is attempting to secure reforms to the UK's relationship with the EU before holding an in-out referendum by the end of 2017.
Ms Eagle, who was greeted with loud cheers as she stood up at the despatch box, said Mr Cameron had been "jetting all over the place" on his "seemingly endless European tour", adding to laughter from Labour ranks: "Can I ask the chancellor: How's it all going?"
"The good news is we have a leader who is respected abroad," replied Mr Osborne, saying the PM was "fighting for a better deal for Britain, something that would never have happened had there been a Labour government".
George Osborne, the bookies' favourite to replace David Cameron as Conservative leader, was then asked whether he really aspired to be "the first post-EU prime minister" by Ms Eagle, who quoted some Eurosceptic Conservative MPs as being "pretty unimpressed by how it's going".
She then imitated Labour leader Jeremy Corbyn's PMQs approach of reading out questions from emails he has received.
In this case it was "Donald from Brussels" - a reference to EU Council president Donald Tusk, who warned of the "destabilising" effect of uncertainty over the UK's future in Europe in a letter to fellow EU leaders this week.
Mr Osborne said the UK had received the "lion's share" of EU investment since announcing its EU referendum plans.
George Osborne went on to say "someone called Tony" has been writing today, referred her to comments from former Labour Prime Minister Tony Blair, who said in an interview published today that the current state of the party was a complete "tragedy".
Ms Eagle, who is shadow business secretary, was ready with an alternative Blair quote: "Just mouth the words five more Tory years and you feel your senses and reasons repulsed by what they've done to our country."
David Cameron is visiting Romania and Poland, so the chancellor - who is PMQs deputy by virtue of also having the First Secretary of State title - stepped up for his second appearance at the despatch box.
Meanwhile Ms Eagle, who is shadow business secretary, was also given the shadow First Secretary of State title in Jeremy Corbyn's first shadow cabinet. This means she gets the job of deputising for her leader on such occasions, and this was her first opportunity since getting the role.
Mr Osborne was given a promotion at the end of the session by Tory backbencher Peter Bone, who called him the "acting prime minister".
Ms Eagle dedicated her opening questions to the flooding that has hit North West England, asking the chancellor to guarantee that money would be "no object" in the relief effort. In his response, Mr Osborne pledged £50m fund for families and businesses that have been affected.
Read more on the £50m flooding announcement
US tycoon and presidential hopeful Donald Trump's call for a temporary halt on Muslims entering the United States did not go unnoticed by MPs, as a petition calling for him to be banned from the UK passed 100,000 signatures.
This means it must be considered for a debate in Parliament. SNP MP Tasmina Ahmed-Sheikh urged Mr Osborne to back the move, while Labour MP Stephen Timms asked him to to respond to Mr Trump's "bizarre claims".
The chancellor said the Republican was "profoundly wrong" but batted away calls for a ban, saying his views should instead be tackled through "robust democratic argument".
Read more about the 'ban Trump' petition
Labour MP Rupa Huq said two curry houses were closing a week due to government policies, and asked Mr Osborne to head off the "coming curry crisis".
It has been claimed that tighter immigration rules are making it harder to hire skilled curry chefs from overseas and Ms Huq said the efforts to train up new chefs in the UK were not succeeding.
We all enjoy "a great British curry" Mr Osborne replied, but said he wanted chefs to be trained in Britain.
"In a funny way it was a bit like PMQs of days gone by" - BBC political editor Laura Kuenssberg
"Feisty Angela Eagle took full advantage of David Cameron's absence... to do that rarest of things - raise Labour spirits" - The Sun
"Angela Eagle had a great debut, getting into some sparky exchanges with Osborne. Better prepped than Corbyn would be" - Asa Bennett, The Daily Telegraph
"George Osborne will be satisfied with that performance, and Angela Eagle will be pleased" - Andrew Sparrow, The Guardian
"This was George Osborne's audition for the top job but he ended up being reduced to a walk on part in the Angela Eagle show" - The Mirror
Labour's Angela Eagle poked fun at David Cameron's EU reforms as she traded jokes with George Osborne at Prime Minister's Questions.