The Spaniard opened the scoring with an impressive free kick 24 minutes in, curling his effort over the wall and beyond the reach of Hugo Lloris.
After Spurs levelled through Michy Batshuayi's own goal, Alonso then struck the winner in the 88th minute, his shot squirming underneath the Spurs goalkeeper's body and into the net.
"I don't think I have struck many free kicks better, not even in training!" he told BBC Sport after the game.
"I am very happy for the win. There was a lot of talk in the week and it showed we are together, we worked hard and we 100 percent deserved the win. I decided to take the free kick and it went well.
"We tried to more of the ball in the second half but they were pressing a lot. We were compact and it was a very good game from us. To get the winner in front of the fans was great."
The last time a Chelsea player bagged a brace away from home in the Premier League was at Leicester City in January, and Alonso was the man who scored those goals.