The Belgian's header from a corner with 10 minutes remaining at an empty Tottenham Stadium sees Spurs go above Arsenal into eighth position in the Premier League table.
With three games left Tottenham have 52 points to Arsenal's 50, but are seven points behind fourth-placed Leicester City.
Victory maintained Mourinho's impressive record in home games against Arsenal -- he has now won six and drawn four -- and was a rare highlight for Tottenham in a mediocre season.
"I could feel the boys' happiness was different. For everyone, the kit man, it was special. We can have two days now to enjoy it but then it's time for Newcastle. It's a ground of course that I have never won at," Mourinho said.
While both teams occupied similar league positions before kickoff, Mikel Arteta's had looked the more impressive of the two since the Premier League resumed.
"I'm disappointed and frustrated and really sorry for the boys. The personality they showed, nullify them and make them hit the ball long but with the two goals we gave away it's very difficult," Arteta said after the match.
The vistors took the lead after 17 minutes when Lacazette struck an unstoppable strike into the top corner from outside of the box.
Spurs levelled three minutes later when Son pounced on a poor back pass from Sead Kolasinac to dink the ball over the goalkeeper.
After the break, Pierre-Emerick Aubameyang almost made it 2-1 but his close range effort struck the bar.
The Arsenal captain was later denied by a Hugo Lloris save before Alderweireld produced the powerful header.
"I think it's massive for us. I think we really wanted to win this game, you know it's difficult without the fans, it will be a totally different atmosphere but I think we did everything that we could. We fought," Alderweireld told Sky Sports.