Albon drove it home cleanly to secure the result, although Ricciardo had hoped the Red Bull driver would make a mistake under the pressure of the situation and hand third place back to him.
The safety vehicle came in at the end of lap six when racing resumed with Bottas thrusting forward to lead, but the contest was brought to a halt immediately after another multiple collision at the back of the field.
The Dutchman had briefly got ahead of polesitter Lewis Hamilton at the Mugello start from third on the grid, before his engine suddenly lost power.
In the wake of Gasly's shock victory for AlphaTauri at the Italian Grand Prix last weekend, the Frenchman has been linked with a potential return to Red Bull.
The 24-year-old British-born Formula one driver finished third behind the Mercedes drivers at the Ferrari-owned circuit.
Speaking after the race, Albon voiced his delight at finally breaking his duck at the end of a tough race. "I am really not happy at the moment, but I can't change it".
"I was in between [Kimi] and Grosjean and I just suddenly got sandwiched in the middle, and we collided in the middle pretty much".
"We did a very good job setting up the vehicle for this weekend compared to some other weekends where we have been a bit further away, where I was not entirely happy with the auto", said Verstappen.
"The team has supported me since Day One". This track is brutal, especially the high speed corners in sector two, and with so many re-starts to manage we had our work cut out. "When we lost two positions on the final re-start, I was sacked up as I wasn't going to let the opportunity of a podium go by so I pushed hard to get third and I'm over the moon with today's result". There's a big difference between thinking you can do it and knowing you can do it.
"We struggled to get off the line so I knew I had to be aggressive to get all the overtakes done but we had a really good auto and we know it's good on the brakes so we used that to our advantage".
"I think actually his feedback is very, very good", Horner explained.
"But also in the back of my mind, I was like, 'Okay, the kid's never been on the podium before".
And carnage resumed again once the safety vehicle pulled in as cars at the back accelerated into the middle of the pack as they tried to gain an advantage upon the restart.