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Cesc Fabregas’ 50th Premier League goal was enough to secure victory for the Blues, who have three matches remaining.
Tottenham play their game in hand against Watford on Monday.
“We got three points and it was very important for us if we wanted to achieve Champions League football,” Conte said.
Fabregas’ curling finish was a fitting way to bring up his half-century of goals in the Premier League, but it was Hazard’s clinical exploitation of a Swansea error that made it, pouncing on Andy King’s sloppy loss of possession, weaving his way forward and picking out the right pass.