The Tigers made it five wins in a row by putting Phil Parkinson’s side to the sword.
Two goals from Kamil Grosicki helped City up to 13th place and to within seven points of Derby County in the final play-off place.
“If anyone talked about the play-offs, they would have the manager all over them,” said Crosby. “It is about one step at a time. We have had a fantastic Christmas and come through with maximum points. But all we do is focus on the next game and that is in the FA Cup (against Millwall).
“We didn’t get too down when near the bottom of the table. We knew we weren’t far away.”
Chris Martin netted his first goal in City colours, while Nouha Dicko, Jarrod Bowen and Evandro also got themselves on the scoresheet. Grosicki, however, was the standout performer for Hull on the day the transfer window opened with speculation suggesting Sporting Lisbon and St Etienne want to sign the Polish international.
“Kamil is working hard for the team and when you get him in the final third he can produce some really good moments,” said Crosby about Grosicki. “He still infuriates us at times with some of his decision-making but he has got undoubted quality.
“His time at the club has been punctured by transfer speculation every single window. Credit to the manager, he has handled Kamil really well. He has motivated him and you can see the fruits of that on the pitch.”
Asked if the window could distract Grosicki, Crosby replied: “I hope not. The owner has already said to the manager that we do not have to sell anyone.”