Ex-Hereford striker Ashikodi missed twice from the penalty spot
Kettering's Jean-Paul Marna scored a late header in extra-time to knock Redditch United out of the FA Cup in the fourth qualifying round.
Moses Ashikodi missed two penalties, hitting the first over the bar in normal time before striking a post in the additional 30 minutes.
Redditch almost won the game three minutes from the end of normal time but James Dance's shot hit a post.
Kettering will travel to Hartlepool in the first round proper.
Redditch assistant manager Micky Moore told BBC Hereford & Worcester:
"It is heart breaking for the club and the players. Football is a cruel game isn't it?
"We took them to the 97th minute on Saturday and the 120th minute today and they have scored in the final 30 seconds of both games. I'd say over the 90 minutes they had the best chance but I thought we played the better football.
"The lads have got to take a lot of credit out of tonight, we have taken a team who are flying high to the last 30 seconds.
"But we have to dust ourselves down, come back training on Thursday and prepare for the game on Saturday against Harrogate."
Kettering manager Mark Cooper told BBC Radio Northampton:
"I can sometimes question [my players'] quality with the ball but nobody can question their commitment.
"With some luck and our fantastic supporters behind us, we've got a chance in two weeks' time.
"People may be knocking Moses tonight but he's made the goal for us this evening. When he's in the team, we've always got a chance of scoring a goal."
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