The England captain was named on Friday as the player behind a gagging injunction involving his private life following a High Court hearing in London.
After the order was lifted newspapers published pages of details about his alleged extra-marital relationship with a "famous" woman.
The controversy did not seem to affect his performance on the pitch, though, and when he was left unmarked at an 82nd minute corner he powerfully headed home the winning goal.
The victory moved the west London side four points clear of champions Manchester United at the top of the table.
Asked about Terry's personal troubles, Chelsea manager Carlo Ancelotti told ESPN: "That is his private life.
"About his work we don't have to say anything. He is a fantastic professional, a fantastic captain, he did a great job for us tonight. All of Chelsea support him and his family."
After 23 matches, Chelsea have 54 points, Manchester United 50 and Arsenal 49, with Arsenal hosting United on Sunday.
Earlier on Saturday, attention was focused on the race for fourth place with current occupants Tottenham Hotspur, and the chasing Liverpool and Aston Villa all in action.
Liverpool and Villa gained ground on Spurs who drew 1-1 at Birmingham City. Liverpool beat Bolton Wanderers 2-0 at Anfield and Aston Villa also closed the gap with a 2-0 win at Fulham.
Spurs stayed fourth with 42 points from 24 games ahead of Liverpool on 41. Villa have 40, with one game in hand, while Manchester City are on 38 with three games in hand.
SUBDUED DISPLAY
Chelsea knew victory at Burnley, who have beaten Manchester Ubnited and drawn with Arsenal at Turf Moor this season, would open up a four-point gap on their closest rivals.
However, their performance was a subdued one until they took the lead in the 27th minute through Nicolas Anelka.
Even then they failed to build on that advantage and allowed Burnley, who have slipped into the relegation zone after a bright start to the season, back into the game early in the second half.
Steven Fletcher out-thought and out-muscled defender Alex before scoring Burnley's first goal since Brian Laws became manager earlier this month.
The joy was short-lived though with Terry's header enough to send Burnley to a fourth straight defeat since he took over.
LIVERPOOL WIN
Across Lancashire a few hours earlier, Liverpool beat Bolton with goals from Dirk Kuyt and an own goal from Kevin Davies.
At St Andrews, Spurs took the lead after 69 minutes with Jermaine Defoe scoring his 17th of the season in all competitions but, just when the victory seemed assured, Liam Ridgewell equalised in stoppage time.
It is the second time in a week that Spurs have allowe
