The Catalan giants seemed set to crash out of Europe's premier club competition at the final 16 stage after a poor first leg performance in Italy left them trailing the Rossoneri 2-0.
However, a double from Lionel Messi and goals from David Villa and Jordi Alba secured a remarkable 4-0 triumph for the hosts at the Nou Camp on Tuesday night and Thompson believes Barcelona - who are now favourites to win the competition - can go all the way after turning the tables on Milan.
"There were a lot of people doubting whether Barcelona could do it but it was a sensational evening and they proved to everyone they're a sensational side," said the former Liverpool man.
"Barcelona only had two shots in the first leg in Milan and this match was a test of their calibre; could they come back and be the first team to overturn a two goal deficit without an away goal? Well they proved they can hack it."
Thompson pointed to a decisive spell just before the interval when Barcelona, 1-0 ahead on the night, profited from M'Baye Niang blowing a one-on-one opportunity before doubling the advantage moments later, as the turning point in the tie.
"Messi got the first goal after five minutes, which was brilliant, but it all hinged on about the 40 minute mark," he said.
"Niang got a clear run through on goal after a Javier Mascherano mistake but he hit the post and it came out.
"One minute later it was in the Milan net and David Villa scored the third with a great finish before Jordi Alba, the left-back, scored in injury time.
"Can they win the Champions League? Yes the can."
Former Liverpool midfielder Dietmar Hamann was equally impressed with Barcelona and said their performance had silenced any critics who had suggested the Spanish outfit were losing their way after back-to-back defeats to Real Madrid in recent weeks.
"That was a short crisis! It's probably over now," the German joked.
"They are a tremendous team but I wasn't sure beforehand, with the way they lost in Milan where they hardly created any scoring opportunities, whether they could do it. But it was a different Barcelona in this match.
"They showed again what they're capable of on their day and you have to have them on your list now for the Champions League. Milan didn't really turn up but Barcelona were magnificent."
Schalke 2-3 Galatasaray (3-4)
In the night's other Champions League round of 16 second leg, Turkey's Galatasaray won an intriguing battle with Schalke.
The German side went ahead through Roman Neustadter to move 2-1 up on aggregate, but Hamit Altintop levelled the tie against his former club before Burak Yilmaz added a second for the visitors moments later. Schalke's Michel Bastos made it 2-2 but, with away goals coming into play and the hosts needing to push forward, sub Umut Bulut capitalised in stoppage time to put Galatasaray into the last eight for the first time since 2001.
Alan McInally: Schalke are really unlucky and they put so much into the second half but that little spell in the first half when they conceded the two goals cost them. They put on a brilliant performance in the second half and I thought at one point they were going to make it. But after they scored their first they didn't really manage to command the next 10-15 minutes and control the game and Galatasaray are dangerous, they have players who can hurt you and that's what they did.
A lot of teams won't want to play Galatasaray after what we've seen here. They've taken a chance on players such as Didier Drogba and Wesley Sneijder - who can unlock doors - and it's going well for them. A lot of teams might seriously think 'we don't want Galatasaray' because it's not an easy place to go.
Hamann: They weren't the two most fancied teams left in the competition but what a show they put on; what fantastic games both legs have been - they were tremendous to watch and a great advert for the Champions League.
I thought Schalke would have enough but I just feel the loss of the injured Klaas-Jan Huntelaar may have cost them the tie because they created numerous openings but lacked the cutting edge, which Galatasaray had. In the end that was probably the difference between the two sides.