Monk had picked up seven points from his first three Sky Bet Championship games in charge, with a Carabao Cup defeat to Everton the only blemish on his record.
But he was dealt a first league loss in the Wednesday hot seat on Tuesday night, as Eaves clinched the three points for the hosts with a superb 72nd-minute header, having come on for Josh Magennis six minutes earlier.
The visitors had arguably had the better of the chances prior to that point, with Barry Bannan testing goalkeeper George Long inside the first few minutes and Sam Winnall heading wide when he should have done better early in the second half.
The result is a first home win since August 10 for the Tigers, who climbed to 12th, three points behind Wednesday.
With only three minutes on the clock, Bannan's curling shot from the edge of the area was tipped around the left-hand post by Long.
Wednesday had opted to leave top scorer Steven Fletcher out after the striker picked up what is thought to be a minor ankle issue in the win against Middlesbrough on Saturday, and the first contribution of replacement Winnall was to head well over from the resulting corner.
Eric Lichaj made a vital interception to prevent an Adam Reach delivery from finding Kadeem Harris in space in the box after 14 minutes, while Winnall felt the full force of Long's clearance of a Bannan cross – the Hull stopper clattering into the forward as he punched clear, prompting half-hearted appeals for a penalty.
Atdhe Nuhiu squandered two promising first-half chances, seeing one blocked by Reece Burke in the box before he failed to get a telling touch on a left-wing cross that glanced across him and away from danger, while Jarrod Bowen saw Hull's best chance of the opening 45 minutes blocked on the edge of the six-yard box before the break.
Winnall headed just wide eight minutes into the second half and Reach had a half-volley from outside the area well held by Long, before Eaves made Wednesday pay.
The striker met Lichaj's delivery with a fantastic stooping header back across Westwood and into the net.
Wednesday's appeals for a penalty fell on deaf ears when Jacob Murphy was brought down on the edge of the box, the referee awarding a free-kick just outside, and Reach sent one last chance over the crossbar in stoppage time to leave Monk's men empty-handed.