Toffees' No. 2 left frustrated after seeing Arteta miss spot-kick before teammate Baines scores his own to share the spoils at the DW Stadium
Mikel Arteta had his penalty saved by Ali Al Habsi in the first half before Leighton Baines was elected to take Everton's second spot-kick, which he converted, handing Everton a point after Charles N'Zogbia had given Roberto Martinez's men the lead before the break.
Round admitted his team felt that they could have claimed a victory in Wigan and help themselves pull away from the clubs trailing behind them.
"We are disappointed.” He told the club’s official website. “We wanted to get to the 50-point mark and put a marker down to the rest trying to catch us.
"Credit to Wigan, they took it to us and if they continue that they have a chance of staying up."
Both Baines and Arteta feature as Everton’s penalty takers and Round is happy to see either player take the spot-kicks, despite the Spaniard’s missed effort against Wigan.
"There is no specific duty," confirmed Round. "Leighton [Baines] and Mikel [Arteta] are our two penalty takers and on any given day either of them can take the penalties. They practice regularly and we have every confidence in both of them."
However, after former Wigan man Baines equalised for David Moyes's side, Round felt confident that Everton would go on and take all the glory.
He added: "They went all out to win, it was important to us to stay in the game, try and control it and take our chances when they came.
"We missed some opportunities, including the penalty, and you think is it going to be one of those days. But credit to the lads they showed great resolve and resilience.
"I thought we could go on and win it but credit to Wigan they didn't lie down."