Newcastle struck three goals in a nine-minute spell as they saw off a spirited display from League Two leaders Luton in the FA Cup third round.
Ayoze Perez pounced on Marek Stech's error for the opener and added a second after Stech had produced a smart save.
Jonjo Shelvey extended the lead, side-footing in his first goal since April, but Danny Hylton steered in after half-time, prompting more Luton pressure.
Lee came off the bench to join his brother Olly in the Luton side as both men tried to peg back the Magpies, for whom their father Rob played for a decade from 1992.
Luton's energy in the second period thrilled a 7,400-strong away support, who thought they were back in the game when Hylton celebrated wildly after finding the net a second time, only for the assistant referee's flag to deny him.
Nathan Jones' side - who have only been outscored by Manchester City in England's top five leagues this season - will rue their first-half capitulation but showed a spirit befitting a side who lead the League Two table by four points.
Jones can now focus on returning Luton to their highest level since 2008 with their push for promotion.
It was perhaps a mark of respect that Newcastle boss Rafa Benitez kept three of his back four in place from Monday's win over Stoke and made just four changes in all after a hectic festive campaign.
Jones' team arrived in the north-east with just one defeat in 14 matches - winning 11 during that run - but Stech's fumble of Gayle's free-kick on 30 minutes gifted the opener and prompted a loss of focus.
Newcastle's midfielders suddenly had more time on the ball, Perez tapped in the second after a fine Stech save denied Dwight Gayle, and Shelvey rolled home to cap a flowing move.
Luton's brief meltdown proved costly but Newcastle ruthlessly seized their chance. Their own intensity then dropped after the break when the away side dug in to make a fight of a tie which, at one point, looked like it could have turned into a rout.
Had Hylton's finish or Lee's strike against the bar given the Hatters a second goal, they could have capitalised on their momentum. But they have now conceded nine goals in three matches and paid the price for a brief lack of resistance on the big stage.
Newcastle - six-time winners of the competition - have now registered back-to-back wins for the first time since September and have three victories in six games.
It would appear they have put the nine-game winless run, which hampered them during November and early December, behind them.
But for a side who have struggled for goals this season, the sight of Perez doubling his tally for the campaign will provide confidence. And they could have scored more, notably through Gayle, who was denied by a fine Stech save later on.