Thomas Cashman, 34, killed Olivia and injured her mother Cheryl Korbel as he chased a fellow drug dealer into their home on the evening of 22 August.
There were gasps and tears in court as he was convicted, with Ms Korbel later saying she felt "ecstatic".
Merseyside Police said Cashman was "not worthy of walking the streets".
Cashman, who was also found guilty of wounding Ms Korbel, the attempted murder of Joseph Nee and possession of firearms with intent to endanger life, will be sentenced on Monday.
The jury of 10 men and two women at Manchester Crown Court took nine hours and three minutes to reach their unanimous verdicts.
Ms Korbel, 46, wearing a pink cardigan and holding a teddy bear, sat with her children Chloe and Ryan in the court as the verdicts were read out.
Cashman wiped away tears in the dock and turned to his family in the public gallery behind, shaking his head.
One of his relatives could be heard saying "appeal it" and they left the courtroom shouting and swearing.
During the trial, the jury heard 36-year-old Nee, who has a number of previous convictions, was the intended target of the attack.
Cashman, who made up to £5,000 per week dealing cannabis in Liverpool, had been lying in wait for his fellow drug dealer at about 22:00 BST, the jury was told.
He shot at Nee in the street and wounded him but his gun jammed as he tried to finish the job.
Nee fled for his life - heading towards the light of an open door - the home of Ms Korbel, who had heard the commotion.
But, as she tried to close the door to keep the strangers out, Cashman shot again.
The bullet went through the door, through her hand, and fatally hit Olivia in the chest.
Cashman then fled the scene, running across back gardens.
The court heard Nee and his family "had their enemies" and it was not the first time he had been targeted in a shooting.
Nee had been shot at on 8 August, two weeks before the shooting in which Olivia was killed.
During a previous hearing, which could not be reported until the conclusion of this trial, the court was told the shootings came after a feud between two families.
There had been a "background of hostility" between Nee's family and another, the court heard.
The same self-loading pistol used by Cashman to kill Olivia had been fired at Nee in the earlier incident, police said.
The defence sought to elicit further material supporting a feud between the two families, including a fight in prison involving two of them and an alleged "straightener" in a pub.
There was insufficient evidence Cashman was involved in the shooting on 8 August but he had not been eliminated by police, the court heard.
During the trial, Cashman, a father-of-two, had told the court he had been at a friend's house where he counted £10,000 in cash and smoked a spliff at around the time of the shooting.
But a woman, who had had a fling with Cashman, told the jury he came to her house after the shooting, where he changed his clothes and she heard him say he had "done Joey".
It can now also be reported that Paul Russell, 41, admitted driving Cashman away from a house where he fled to following Olivia's murder and disposing of his clothing.
Russell, of Snowberry Road, Liverpool, pleaded guilty to assisting an offender at a hearing at Liverpool Crown Court in October and will also be sentenced on Monday.
Journalists were prevented from reporting his plea until the conclusion of Cashman's trial.
Det Supt Mark Baker, the senior investigating officer in the case, said Cashman's actions were "abhorrent".
He said officers were still "hunting down" those who had enabled Olivia's murder and finding the weapons, which had not yet been recovered, was key.
"When he found out that he had shot an innocent young girl, he should have had the courage to stand up and come forward," he said.
"Instead, he chose to lie low despite the fact that he was a dad himself.
"He is not worthy of walking the streets of Merseyside, and neither are those who think they can bring fear or intimidation to our communities through the use of firearms."