Hull are one round away from the Europa League group stages after Sone Aluko gave them victory over AS Trencin.
The FA Cup runners-up suffered an early setback at the KC Stadium when Tomas Malec scored for the Slovaks after 91 seconds.
Ahmed Elmohamady equalised before the half hour in the third qualifying round tie. And with 10 minutes remaining, sub Aluko fired home from close range.
Hull now go into Friday's draw for the Europa League play-off round.
The Tigers, who had never played in a continental competition before qualifying as FA Cup runners-up, needed to win, or else go to penalties, to qualify after a goalless first leg.
A Trencin goal inside two minutes meant Steve Bruce's side needed to score at least twice to remain in the competition.
Malec controlled a low cross to sweep the ball beyond goalkeeper Allan McGregor.
Hull drew level in the 27th minute when Robbie Brady found the space to pick out Elmohamady, whose powerful header squeezed over the line.
Despite pressing for a second after the break, Hull were in danger of running out of time as they fought to avoid a premature end to the club's first European adventure.
The breakthrough finally arrived in the 80th minute, home debutant Robert Snodgrass heading Elmohamady's cross into the six-yard box for Aluko to prod home.
Hull manager Steve Bruce: "You're always worried when there's 15 minutes to go and you're thinking about all the hype and euphoria, but you're about to fall at the first hurdle.
"But we've stuck at it and, not for the first time, we've seen them do it. It's been difficult but we've got through it and let's enjoy the draw tomorrow."
- 4BruceSubstituted forInceat 66'minutes
- 27El Mohamady
- 11BradySubstituted forRoseniorat 56'minutes
- 9LongBooked at 39minsSubstituted forAlukoat 74'minutes
- 22Rodriguez da Silva
- 14HolubekBooked at 63mins
- 23HajradinovicSubstituted forOpatovskyat 74'minutes
- 27SimonSubstituted forRundicat 90+1'minutes
- 6MalecBooked at 90mins
- 7MisakSubstituted forMondekat 82'minutes
- Bas Nijhuis
- Shots on Target