Striker Abel Hernandez struck in either half as Hull hit back from a wretched start, having conceded an early penalty which was converted by Junior Stanislas.
Bournemouth defender Tyrone Mings' own goal after the break gave Hull a two-goal cushion and, in Silva's first league game in charge, they won for the first time in 10 top-flight matches to climb off the foot of the table.
But former Sporting Lisbon and Olympiacos boss Silva was already looking forward to another busy week on the training field and in the transfer market.
"I'm happy, but we have a lot of work to do in the next few weeks," the Portuguese said.
"We'll try to find solutions for our squad because we need to improve
I'm happy because we need to win games and it was my first game in the Premier League
"In one week we've played three games and we've won two games - it's fantastic because in the last two months we didn't win one game.
"I believe in our team and our quality
The players are happy
They're starting to believe and understand better what I want for the team."
Silva was careful not to single out only two-goal hero Hernandez for praise after the Uruguay international scored his third goal in as many appearances since returning from a two-month lay-off due to a hernia injury.
"First of all it's a good day for the team and a good day for our fans," Silva added
"There was a fantastic atmosphere in our stadium, fantastic support
This is very important for us.
"I said to the press in pre-match that we need them (the fans), we need the atmosphere here because we are stronger with them.
"About Hernandez - it's really important for a striker when they score goals.
"In the last two games at home Hernandez has scored three goals and that's important for the team and for him also
His second goal was a very good goal."
Bournemouth boss Eddie Howe was disappointed his side failed to capitalise on their flying start after Hull defender Harry Maguire had bundled over Ryan Fraser in the area and Stanislas buried the third-minute spot-kick
The Cherries had dominated most of the first half, but Hernandez headed home Andrew Robertson's inviting cross at the far post in the 32nd minute to spark the Hull comeback
"We were dominant for the first half-hour - it was the perfect start for us
We were in control of the game," Howe said
"I just felt we didn't really go for the throat
I felt we were content with one (goal) and the flow of the game changed when they equalised
"From that moment we never really got control back."
Hernandez scored a superb second in the 50th minute after cutting inside on the right edge of the area and Mings diverted Tom Huddlestone's shot into his own net 12 minutes later
"Today, 1-0 up away from home, you expect to be more solid, to see the game out in a professional way," Howe added
"I don't think we created a huge amount of chances and we got punished and that's something that has to change."
Source : PA