Ricardo Carvalho's sixth-minute header sealed the victory for the Blues, who leapfrog Arsenal into second place in the Barclays Premier League table.
The hosts dominated possession in the remainder of the match but were unable to convert it into goals and the visitors almost stole a share of the points, hitting the woodwork three times in the closing stages.
Terry told Setanta Sports 1: "It was not a great performance today - we didn't start very well and there were a few tired legs out there, but we got the three points and held on.
"Middlesbrough have some big players and we attacked the ball well in defence and managed to deal with the late pressure.
"We dug in well today and this lifts us up to second which is good. All we can do as a team is keep the pressure up on Manchester United and carry on winning."