Second half goals from Francesco Cozza (34) and Franco Brienza (53) cancelled out Inter's early two-goal lead through Douglas Maicon and Patrick Vieira on 9 and 24 minutes.
After two goalless draws against Genoa and Fiorentina, The Nerrazuri were keen to score and early goal and they nearly went in front when Ricardo Quaresma hit the upright.
On 9 minutes, Maicon scored a brilliant shot from outside the box after being set up by Zlatan Ibrahimovic.
Fifteen minutes later, Patrick Vieira marked his comeback from injury by scoring from a low drive from outside the Reggina box.
The home side started their comeback route in the 34th minute courtesy of a powerful Cozza shot that left Julio Cesar rooted to the ground.
In the second half, Edgar Barreto wrapped his right down the right flank by crossing to Brienza and the latter took full advantage by slotting the ball past Cesar.
Inter saved the day for Jose Mourinho through the unlikely source of Ivan Cordoba who picked up the lose ball from a Quaresma corner and fired a low drive past the Reggina defense.
Reggina stay bottom of the table with five points, while Inter move to first with one game over second place Udinese.