The Blues were pushed all the way to the wire in their Premier League opener, after James Collins' equaliser had cancelled out Eden Hazard's penalty.
However, they dug deep and got the result they were looking for in the closing stages of the contest as Diego Costa slotted home from outside the box to ensure manager Antonio Conte made a winning start at the Stamfod Bridge helm.
Terry highlighted the importance of the opening-day win, which he felt was crucial after a disappointing 2015/16 campaign.
"It is very important for us to get off to get off to a good start. We did not start off or finish well last year. That is the fight and commitment we want. We dug in in the London derby and that is great for the management," he told Sky Sports.
"Eden Hazard is up there with the best in the world. We relied on him too much last year, he cannot do it on his own.
"To ourselves, to our supporters and to our friends and family, we will slowly get there. The consistency is important, the winning start was vital."
Hazard, who claimed the man-of-the-match award, added: "It was good to be back on the pitch in front of our own supporters. We did very well and we played 90 minutes like tigers.
"It's not every year I score in my first game. Last season is finished. We are looking forward to the future. We want to bring a lot of emotion to the fans."