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In eight-time winners Tottenham's first FA Cup tie at their new stadium, Lo Celso slotted them in front after two minutes before Lamela doubled the lead in the 15th minute.
Second-tier Middlesbrough made it an anxious final few minutes for the hosts as George Saville's low shot halved the deficit in the 83rd minute but Tottenham survived for a first win in five games to set up a fourth-round trip to Southampton.
Newcastle United strolled past third-tier Rochdale, winning 4-1 at St James' Park. Eoghan O'Connell's own goal put Newcastle ahead and Matthew Longstaff and Miguel Almiron ended any chance of an upset with goals before halftime.
Joelinton ended his 20-game goal drought in the second half as Newcastle cruised through to face third-tier Oxford United.
Third-tier Shrewsbury Town set up a dream fourth-round home tie against Liverpool as they sprang something of a surprise by knocking out Championship club Bristol City 1-0.
Defender Aaron Pierre's superb long-range effort in the 89th minute sparked wild celebrations at New Meadow.
"Football's all about entertainment, enjoyment -- the FA Cup is special -- and now we get the chance to play Liverpool," Shrewsbury boss Sam Ricketts said.
Former winners Coventry City, who have fallen on hard times, beat Bristol Rovers 3-0 in their replay to set up a 'home' tie with Birmingham City at Birmingham's St Andrews ground where they are currently tenants.
Second tier Reading won 2-0 at Blackpool.
Tranmere Rovers' replay at home to top-flight Watford was postponed after heavy rain left the pitch waterlogged.
Mourinho made five changes to his starting lineup with Lo Celso and Lamela, who both impressed as substitutes in the 1-0 defeat by Liverpool on Saturday, given the chance to impress.
They did just that although both goals came from poor Boro defending. The 4,000 Boro fans who made the trip south were silenced after two minutes when keeper Tomas Mejias played a risky pass out and Lo Celso intercepted before cutting inside and side-footing home.
Boro almost levelled when Lukas Nmecha left Japhet Tanganga on the ground after a great turn but he was denied by Paulo Gazzaniga who had a relatively quiet evening.
Tottenham's second goal was almost a carbon copy with Lamela seizing on some sloppy Boro play before slicing through and finishing neatly with the outside of his left foot.
Mourinho's side failed to turn their domination into more goals and things got edgy late on when Saville's shot evaded Gazzaniga who seemed slow to get down.
"I told my players at halftime if we don't make it 3-0 then if 2-1 we would be in trouble and it happened," Mourinho, who has gone on a nine-match run without a clean sheet for only the second time in his 923-game managerial career.
"We did lots of things well, we conceded the goal, a bit frustrating, but more frustrating was that we did not score three, four or five."