The striker was among Coventry's most dangerous players at the Coventry Building Society Arena but could not turn his opportunities into goals as Hull secured a late victory.
Awoniyi, who joined Coventry from Nottingham Forest in a nine-million-pound deal earlier this month, had several chances to open his Premier League account for the club.
Hull goalkeeper Konstantinos Tzolakis stood in his way, making a number of important saves to frustrate the former Forest striker. One of the clearest opportunities fell to Awoniyi from close range, but he failed to make the decisive connection.
Coventry dominated possession for long periods and looked capable of securing their first league points of the campaign. Instead, they were caught on the counter-attack eight minutes from time, with Liam Millar finishing to give Hull all three points.
Awoniyi did not hide from the disappointment and accepted that Coventry needed to take greater responsibility for failing to capitalise on their chances.
"We did quite well in terms of having the ball, but unfortunately, they had that great counterattack, and they took their chance," Awoniyi said, as quoted by Pulse Sports Nigeria.
"It is on us to take our chances as well."
The striker also acknowledged Tzolakis' contribution, describing the Hull goalkeeper as brilliant after his impressive display.
Although his league campaign has yet to produce a goal, Awoniyi has already found the net for his new club, scoring in Coventry's 4-2 Carabao Cup victory over Plymouth Argyle.
With Coventry facing Manchester City next on September 5, Awoniyi will be desperate to turn his promising performances into goals and help Lampard's side finally get their Premier League campaign off the mark.