Sparta levelled the tie in the 17th minute through David Lafata and the hosts were frustrated by wasteful finishing and stubborn defending until Hazard pounced in the last minute at Stamford Bridge.
When asked about Chelsea's clinical edge, a relieved Terry told ESPN: "We showed that in the last minute tonight.
"We kept going and going and going. The first 80 minutes nothing went for us and really we had a lot of attempts on goal and didn't get it, but the persistence from the lads was fantastic and you know credit to us.
Terry revealed that the European champions were spurred on to reach the last 16 after watching the Champions League games on television.
"It was an absolute must for us," he said. "We're very disappointed to be out of the Champions League and watching it you know Wednesday and Tuesday nights is disappointing, but that's enough motivation for us to kick on in this competition.
"We are in it to win it and nothing changes from that. After that we've got a chance with the Super Cup. There's still some big competitions to play for."
Describing Hazard's late heroics, Terry said: "Great goal from Eden. The manager rested a couple tonight and you know the likes of Eden on the bench, he's great to have when he comes on.
"He showed great sharpness and got the goal to top it off."