Babina Noko were knocked out in the quarter-finals after suffering a 3-2 defeat on penalties at the Peter Mokaba Stadium. The two sides had played out a 2-2 draw after extra time, forcing the match to be decided from the spot.
Sekhukhune's elimination came on a day that saw another Betway Premiership side fall to lower-division opposition, as AmaZulu FC were stunned by Casric Stars earlier in the day.
For Sekhukhune, the defeat represents a significant blow to their hopes of ending the season with silverware. The club invested heavily in strengthening the squad at the start of the campaign with the ambition of competing for trophies.
Despite the disappointment, Tinkler insisted he was proud of his players' display and felt their performance deserved a better outcome.
"I am very happy with the performance that our boys put on tonight," Tinkler told journalists after the match.
"I think they were exceptional and I don't think we've played like that since the break. So I'm very proud of the effort they put in. It's heartbreaking for them, but I'm very proud of their performance."
The former Bafana Bafana midfielder was also quick to reject claims that his side took Milford lightly.
"No, not at all. Like I said, our team performed exceptionally well," he said.
"We were by far the best team and we created the most chances during the majority of the game. They spent the match defending, so I don't think we underestimated them.
"We all knew that this was never going to be an easy match against Milford. It doesn't matter whether they are a division lower or not. You always have to raise your level and play to a higher level, and I thought we did that today."