Ricardo Carvalho's sixth-minute goal ended up being the decider but Boro hit the woodwork three times and might also have had a penalty.
The win took Chelsea above Arsenal into second place in the Barclays Premier League table and five points behind leaders Manchester United - who travel to Boro next week.
Grant conceded that his players had been forced to hang on at the end but correctly countered that more clinical finishing earlier on, especially from Didier Drogba and substitute Shaun Wright-Phillips, could have put the game out of the visitors' reach.
He said: "They hit the crossbar two times but at the other end Shaun could have scored twice and Didier in the first half.
"I cannot say that it was our best game but it was important for us to win today rather than have quality in the game.
"It is a good win. They were unlucky, but we also created some good chances from open play. We started very well and, maybe because of that, we thought it would be easy.
"We controlled the ball and controlled the game but by not making it 2-0, it became a good chance for the other team. But I think we deserved the win. At this stage of the season, it's about winning."