Abel Hernandez was the hero for the hosts, scoring twice on his first start since injuring his groin against Southampton on 6 November.
That match against the Saints was the final league victory under Silva's predecessor Mike Phelan, who was dismissed on 3 January with the team bottom of the league.
Hull have now won twice in three games under former Olympiakos and Sporting Lisbon boss Silva, having beaten Swansea in the FA Cup third round a week ago.
Tyrone Mings' own goal added gloss to the scoreline for Hull, who took a long time to grow into the match after falling behind to Junior Stanislas' third-minute penalty, awarded for Harry Maguire's challenge on Ryan Fraser and the 10th the Tigers have conceded in 21 games this season.
Bournemouth have now shipped three goals in eight of their last 10 games in all competitions, and drop to 11th place after their 10th defeat of the campaign.
It has been a bleak winter for Hull so far and the loss of £10m striker Hernandez to injury certainly contributed to their plight.
The Tigers went nine games without a win during his absence, dropping them to the bottom of the table by Christmas and culminating in the departure of Phelan early in the new year.
His return has coincided with the appointment of Silva, and it has quickly become apparent that the influence of both could prove vital in turning Hull's season around.
The 26-year-old's goal against Swansea in the cup was a timely reminder of the ability that brought him 20 goals in Hull's promotion campaign last season, and his first against Bournemouth bore all the predatory instincts that the hosts have been missing these past few weeks.
Huge credit must go to Andrew Robertson for the quality of his cross from the left wing, which presented Hernandez with a headed chance from around a yard out that he simply could not miss.
The second was more of a self-made effort, the Uruguay international picking the ball up on the right, advancing into the area and selling a couple of cute stepovers to Mings before curling a precise finish into the left-hand corner.
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It was a huge result for Hull but also a significant one for the visitors, who have now conceded 24 goals in their last 10 league and cup games.
Their mid-table position - and nine-point gap to the drop zone - means Bournemouth have some wiggle room as they bid to shore up at the back.
But they cannot continue to leak goals at the present rate and not expect to find themselves flirting with danger at the end of the season.
Hull's third goal here was certainly unfortunate - Mings diverting in Tom Huddlestone's hopeful shot from the edge of the area - but nevertheless continues a sequence of results that is clearly worrying boss Eddie Howe, who conceded that there was a "pattern emerging".
Hull head coach Marco Silva: "I'm happy because what I saw was a team with big spirit and big character. That's important for me.
"After 10-15 minutes it was difficult, but we started to control the game and put the other team back. The first 25-30 minutes of the second half it was all our team.
"I start to see [progress]. We have time next week to work more and the players now believe more in our ideas.
"I saw my players smile in the dressing room and that's important for me."
Bournemouth boss Eddie Howe: "We got the perfect start and from that point we were looking to consolidate and then it slipped away. The second half was tough to watch.
"The frustrating thing is we didn't go for the game to try and put it to bed.
"We're disappointed with all three goals. We're in the middle of the season and we have everything to play for, Our season could still go two ways."
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Hull are back in Premier League action on Sunday, 22 January with a tough trip to Stamford Bridge to face leaders Chelsea (16:30 GMT), while Bournemouth host Watford on Saturday, 21 January (15:00 GMT).
Match ends, Hull City 3, Bournemouth 1.
Second Half ends, Hull City 3, Bournemouth 1.
Jack Wilshere (Bournemouth) wins a free kick on the left wing.
Foul by Evandro (Hull City).
Offside, Hull City. Oumar Niasse tries a through ball, but Adama Diomande is caught offside.
Offside, Bournemouth. Steve Cook tries a through ball, but Callum Wilson is caught offside.
Marc Pugh (Bournemouth) wins a free kick in the defensive half.
Foul by Harry Maguire (Hull City).
Joshua King (Bournemouth) wins a free kick in the defensive half.
Foul by Andrew Robertson (Hull City).
Offside, Hull City. Jake Livermore tries a through ball, but Oumar Niasse is caught offside.
Substitution, Hull City. Jake Livermore replaces Ryan Mason.
Harry Arter (Bournemouth) wins a free kick in the defensive half.
Foul by Evandro (Hull City).
Substitution, Hull City. Oumar Niasse replaces Abel Hernández.
Attempt saved. Abel Hernández (Hull City) right footed shot from the left side of the box is saved in the bottom right corner. Assisted by Tom Huddlestone.
Attempt missed. Harry Maguire (Hull City) right footed shot from outside the box misses to the left. Assisted by Adama Diomande.
Foul by Adam Smith (Bournemouth).
Sam Clucas (Hull City) wins a free kick on the left wing.
Substitution, Hull City. Evandro replaces Robert Snodgrass.
Adam Smith (Bournemouth) wins a free kick in the defensive half.
Foul by Curtis Davies (Hull City).
Foul by Tyrone Mings (Bournemouth).
Robert Snodgrass (Hull City) wins a free kick on the right wing.
Andrew Surman (Bournemouth) wins a free kick on the left wing.
Foul by Robert Snodgrass (Hull City).
Delay over. They are ready to continue.
Delay in match Tyrone Mings (Bournemouth) because of an injury.
Corner,  Bournemouth. Conceded by Eldin Jakupovic.
Attempt saved. Jack Wilshere (Bournemouth) left footed shot from the centre of the box is saved in the top left corner. Assisted by Marc Pugh.
Attempt saved. Jack Wilshere (Bournemouth) left footed shot from outside the box is saved in the centre of the goal. Assisted by Andrew Surman.
Substitution, Bournemouth. Marc Pugh replaces Junior Stanislas.
Charlie Daniels (Bournemouth) wins a free kick in the defensive half.
Foul by Adama Diomande (Hull City).
Adam Smith (Bournemouth) wins a free kick in the defensive half.
Foul by Andrew Robertson (Hull City).
Offside, Bournemouth. Steve Cook tries a through ball, but Junior Stanislas is caught offside.
Jack Wilshere (Bournemouth) wins a free kick in the defensive half.
Foul by Harry Maguire (Hull City).
Own Goal by Tyrone Mings, Bournemouth.  Hull City 3, Bournemouth 1.
New boss Marco Silva guided Hull City to victory in his first Premier League match as the Tigers moved off the foot of the table with an impressive performance against Bournemouth.