The Argentinian striker insisted on taking the spot-kick despite fierce protests from team-mate Robert Snodgrass, who apparently expected to be given the responsibility, and the opportunity was wasted as Becchio drove a rash finish high over the crossbar.
United were leading by four goals at the time and Becchio's miss did not affect the outcome of a one-sided League One fixture, but Grayson has vowed to avoid a repeat of that scenario in the final weeks of this term.
The Leeds boss admitted concern at the club's poor record with penalties, and United have failed to convert their last four chances from the spot.
Becchio was previously denied during United's 2-0 win at Brighton in January while Jermaine Beckford and Lee Trundle have also been unsuccessful with attempts from 12 yards.
Winger Andy Robinson has been a reliable penalty taker - the former Swansea City captain found the net against both Bradford City and Northampton Town - and Grayson will insist on greater discipline and composure among his players when future opportunities arise.
Grayson said: "We will have a recognised and designated penalty taker, and if he's on the pitch and wants to take it then he will take it.
"It's an area that we need to address and one you can overlook at times. It's a concern because missing penalties from 12 yards is something we shouldn't be doing. It can cost you games in the long run.
"We found that out when (Trundle) missed one at Hereford because if we'd scored that night then I think we'd have won the game. The other night didn't matter so much but you still have to have a ruthless streak.
"The players are aware of it now and I don't think you'll find anyone pulling or pushing for the ball."