It was confirmed on Thursday that Racing Point, which will morph into Aston Martin from the 2021 season, had replaced Sergio Perez with Vettel for at least next season.
Vettel, a four-time champion, finished fifth in the F1® championship last season but looks set to get nowhere near that this time around, with Ferrari enduring a miserable campaign.
Ferrari announced earlier this year that the German's contract was not being renewed, sparking speculation as to his next move.
He revealed in Thursday's drivers' news conference ahead of the Tuscan Grand Prix that he had even considered retirement, but Hamilton – whom Vettel was once a close rival of – is happy to see his old adversary get a move.
"I thought it was going to happen, so I'm glad to hear it is over," Hamilton said of Vettel's move. "He's going through a rough patch at Ferrari, but I think at Racing Point [Aston Martin] he's in the right place. Direction can help.
"It's a shame for Sergio [Perez], but I think he has other options.
"Honestly, I thought that was the ideal direction that [Vettel] was going to go. I thought that it would happen. Maybe I'm biased, but I believe that experience counts for a huge amount.
"He's obviously had a difficult time at Ferrari, but he's a four-time champion that can steer that team even further in a better direction in terms of car development.
"I think you should never take that for granted. It's a great move for the team."
The weekend's meeting at Mugello will be Ferrari's 1,000th grand prix and be treated as something of a celebration of the iconic team, with the safety cars donning red livery.
While few are expecting them to offer any kind of challenge at the top end, Hamilton paid a brief tribute to Ferrari ahead of the race.
"F1® wouldn't be the same without Ferrari," he said. "Hopefully they will re-join the fray soon."
Team-mate Valtteri Bottas added: "One thousand races is an impressive number. They are having a tough time at the moment, but they will certainly fight back."