Coventry started the brighter of the two sides with Dele Adebola and Michael Doyle both going close in the first four minutes.
But Hull, who started the game on the back of three consecutive losses, started to take a grip on the game and almost took the lead in the 17th minute.
Captain Nicky Barmby played a ball through to Chris Brown who beat the offside trap but saw his shot blocked by the on-rushing Stephen Bywater.
From the resultant throw-in, the Tigers did take the lead as Welsh turned and shot, with the ball beating Bywater thanks to a deflection from the unfortunate Richard Duffy.
The goal visibly gave Hull confidence and they went on to dominate the rest of the half. They should have doubled their lead in the 27th minute when on-loan Sunderland striker Brown narrowly failed to get on the end of a Nicky Barmby free-kick.
Brown should have had a first-half hat-trick as he blazed over three minutes later and then shot wide after catching Stephen Hughes in possession.
Coventry almost levelled in the 38th minute after lovely inter-play between James Scowcroft and Adebola saw the for the Birmingham striker cross for McSheffrey who was denied by a brilliant diving save from Myhill.
The Sky Blues again started the second half the better with Adebola heading over from a Paul Watson corner in the 50th minute.
Five minutes later the same combination saw Adebola's header on target this time - only to be cleared off the line by Welsh.
Desperately seeking an equaliser, Adebola cannoned a header off the crossbar after a lovely flighted cross by substitute Willo Flood.
Flood's presence boosted Coventry and Mark Lynch was sent off for a second bookable offence after a cynical foul on the Irishman.
But against the run of play, Hull doubled their lead with a real sucker punch.
Substitute Andy Morrell was caught in possession by Welsh and his run ended with a sublime curling shot that Bywater could do nothing about.
The goal killed the game as a contest as Hull were prepared to put men behind the ball and soak up Coventry pressure, with Scowcroft and teenager Liam Davis going close on his debut, after replacing McSheffrey in the 81st minute.