Stephane Sessegnon scored the only goal of the game just before half time to earn the Black Cats their first victory over the Toffees since 2001.
Di Canio told BBC Sport: "We did a good job. The lads have done an amazing job in the last few weeks and this is the result. I was a footballer at many clubs and it isn't easy when there's a change. The players can smile now.
"But 37 points are not enough. If Wigan win their game in hand, they'll be on 34 with four games to go. Today was a step forward but the main job is not done."
Sunderland are 14th in the Premier League and are six points clear of the relegation zone with four games to play this season.