You got the verb right. Sveglia would be the old clock with the bells that ring, but as technology evolved we kept the same word for both the clock and waking-up alarms, so you can say "Spegnere la sveglia" whether you turn off the alarm on your phone, or digital clock, etc. To describe the ringing of any alarm set to wake you up you can say "è suonata la sveglia (e l'ho spenta)". But if it was an alarm set for any other event (reminders, cooking tasks), matters would be complicated, as you could also say "è suonato l'allarme (e l'ho spento)/si è spento".

Berze is right I would add that allarme would be also the allarm like those ones: burglar alarm antiaircraft. Spegnere is to switch off in English as you correctly said ciaooo Ale

Turn off or switch off means spegnere a electronic devices in Italian.