Jose Mourinho's side almost killed off second-tier Middlesbrough within the first 15 minutes of the third round replay at the Tottenham Hotspur Stadium.
Lo Celso, on loan from Real Betis, got Tottenham up and running with only his second goal of the season and his first since a Champions League strike against Red Star Belgrade in November.
It was an encouraging moment for the Argentine midfielder, who has endured a difficult season in north London.
Tottenham have not lifted a major trophy since 2008 when current Middlesbrough manager Jonathan Woodgate scored the League Cup final winner against Chelsea.
They needed a Lucas Moura equaliser at the Riverside Stadium to avoid a shock exit in the 1-1 draw at Middlesbrough earlier this month.
But Mourinho still made five changes from last weekend's defeat against Liverpool, with Christian Eriksen, linked with a January move to Inter Milan, starting despite the Dane's form being affected by the transfer talk.
A gift from Middlesbrough keeper Tomas Mejias helped Tottenham open the scoring in the second minute.
Mejias tried to play a quick ball out and ended up passing straight to Lo Celso, who ran across the area before firing into the bottom corner.
Tottenham's second came when Argentina winger Erik Lamela sliced through the Middlesbrough defence before slotting home in the 15th minute.
George Saville got one back for Middlesbrough in the 83rd minute as Tottenham survived a tense finale to earn a trip to Southampton.
Joelinton ends drought
At St James' Park, Newcastle wasted little time putting third-tier Rochdale to the sword as they booked a home date with League One side Oxford in the last 32 with a 4-1 win.
Magpies boss Steve Bruce had declared his determination not to lose to "bloody Rochdale" as he eyes a sustained Cup run for a club that has not won the competition since 1955 and last reached the final in 1999.
But Newcastle's FA Cup form is so bad that they have not gone beyond the fourth round for 13 years.
They were never in danger of an embarrassing exit, in front of controversial owner Mike Ashley, after Rochdale defender Eoghan O'Connell turned Matt Ritchie's cross into his own net in the 17th minute.
Matty Longstaff doubled Newcastle's lead with a clinical close-range finish three minutes later.
Miguel Almiron had put Newcastle in front in the 1-1 draw at Rochdale and he bagged the Magpies' third goal this time, seizing on a misplaced pass from keeper Robert Sanchez and slotting home in the 26th minute.
In the 82nd minute, the beleaguered Joelinton converted a cross from debutant Tom Allan and showed his relief as he sprinted to hug Bruce on the touchline.
That ended the Brazilian striker's 20-game goal drought and brought him a first goal at St James' Park since his move from Hoffenheim last year.
Third-tier Shrewsbury earned a home tie against Premier League leaders Liverpool as Aaron Pierre's 89th-minute strike clinched a surprise 1-0 win against Championship side Bristol City.
Coventry's 3-0 win against Bristol Rovers set up the unusual prospect of the League One side being the 'home' team against Birmingham in the fourth round -- even though they are currently tenants of their local rivals at St Andrew's.
Watford's trip to Tranmere was postponed due to a waterlogged pitch caused by Storm Brendan.