The relegation-threatened Latics looked to be on course for three much-needed Barclays Premier League points at the DW Stadium when Hugo Rodallega needlessly handled from a corner.
That allowed former Wigan defender Leighton Baines the opportunity to step up and smash an equaliser from the spot after 77 minutes.
Wigan had led courtesy of a fine Charles N'Zogbia strike in the 21st minute and already survived one penalty scare when Ali Al Habsi saved from Mikel Arteta.
Round said: "It was a struggle. We were playing against a team that are fighting for their lives. It was always going to be tough; they were always going to go all out to win.
"It was important we tried to stay in the game, control it and take our chances when they came. Unfortunately, we missed a couple of chances and missed a penalty and we thought it was going to be one of those days. But credit to the lads, they showed great resolve and resilience, as Everton do."
Source: PA
Source: PA