The home side rose back up the table thanks their 2-0 win, with an own goal breaking the deadlock before January recruit Cenk Tosun got his first goal at Goodison Park.
With his side trailing 2-0 in the 88th minute, Ryan dived well to his right to prevent further damage from Rooney's well-hit spotkick.
Things could have been worse for Brighton after Anthony Knockaert's dismissal but Wayne Rooney's penalty was excellently saved, and it proved to be the only positive on an afternoon that Hughton told his club's official site he was ready to forget immediately.
"The second goal was a nice little lay-off to Cenk and a handsome finish and then the skill into the box from Dominic [Calvert-Lewin] to get the penalty - I don't think anybody's got any doubt about that, it's a brilliant piece of skill to get around the defender in that tight situation".
"It's a shame Wayne missed the penalty because he had a very good performance in a sitting midfield role". "His passes were the reason we were so fluid I felt in the fist half and then again the second half".
FT. Clean sheet + two goals = three points.
And of the Seagulls' performance, he said: "I was surprised, particularly with the form we have been in". Brighton, who had Anthony Knockaert sent off for a unsafe tackle, dropped to 11th after a first defeat in eight in all competitions, now below Everton, whose ninth Premier League home win of the season lifted them to ninth.
"At half-time, having not played well, we were still very much in the game".
"We restricted them to minimal chances going into halftime".
Walcott scored twice against Leicester in just his second game for Everton but has failed to net in his subsequent five matches, and saw an effort ruled as an own goal on Saturday.