Adam Johnson became the second Sunderland player to score in the Premier League this season, but Everton struck twice in the space of two minutes late in the second half to extend their unbeaten run in the competition to eight games and consolidate fourth place in the table.
Martin O'Neill's team finally ended a goal drought that stretched back eight hours and 55 minutes, which was deserved reward for an improved performance, but they were undone when Marouane Fellaini and Nikica Jelavic transformed Everton's deficit into profit in the 77th and 79th minutes. Improved or not, Sunderland have now won once in 10 attempts in the Premier League this season.
In personal terms, this was a meeting of two of the best managers never to have had a club big enough to challenge for the title. Form is transitory, class permanent, but, at the moment, David Moyes is ahead on points, literally and metaphorically. With Everton he has done wonders on a pauper's budget, mastering the alchemist's art to keep his charges competitive while selling top players, such as Jack Rodwell, who left for £12m before the season started.
His eye for a bargain is typified by Jelavic, who – bought from Rangers for a modest £5.5m in January – was an immediate hit in England, with nine goals in his first 13 league appearances last season and five in 10 this time. The versatile, endlessly influential Fellaini has been an even better buy and has six goals in his past 10 games.
O'Neill's reputation is such that an immediate transformation was expected when he replaced Steve Bruce at Sunderland 12 months ago. It has not been forthcoming and, a year into his tenure, the revivalist Ulsterman finds himself at the wrong end of the table, his team averaging less than a point per game, which is relegation form.
Their problem is easy to identify. Before Saturday, only Steven Fletcher had scored in the league and his five goals have been nowhere near sufficient. No team has scored fewer goals or had fewer shots.
Sunderland had the better start and Fletcher ought to have scored after seven minutes, but he shot wide across the face of the goal, from left to right. Substituted later with hamstring trouble, he is expected to miss Scotland's friendly against Luxembourg on Wednesday. Stéphane Sessègnon also had a decent early chance and O'Neill felt his team should have been 2-0 up after 10 minutes and "out of sight" by half-time.
For Everton, Steven Pienaar, Jelavic, and Phil Neville tried their luck in the first half to no avail. Sunderland, meanwhile, were assembling one of their best performances of the season and, as the first half went into added time, they took the lead after a corner from Sebastian Larsson that Everton failed to clear. Craig Gardner was therefore allowed to set up Johnson, who beat the advancing keeper, Tim Howard, to the ball to slot it into the vacated net.
Early in the second half, Johnson added to his notable contribution with a goalline clearance from John Heitinga.
Their passing triangles having got them nowhere, Everton decided to try the longer ball, to no avail until Fellaini, the man of the match, drilled a low shot into the right-hand corner of Simon Mignolet's goal.
For Everton's winner, Jelavic hit the Sunderland net from six yards out after Fellaini had flicked through a pass from Leon Osman.
Moyes said: "I thought it was going to be one of those days, and Sunderland were going to win 1-0, but we kept at it, and in the end defeat would have been an injustice given all the possession we had. Their resilience made it hard for us to score and we were a bit stodgy at times. It didn't look like it was coming, but maybe the amount of pressure we had in the second half paid off."
Unsurprisingly, O'Neill saw it differently. "Our play in the first half was exhilarating and we should have been out of sight at half-time," he said. "We had two clear chances before the goal and, instead of 1-0, we should have been three up.
"Obviously I'm very disappointed that we didn't see it through, but it was a good performance against a side with a great chance of being in the Champions League next season. The players are encouraged and we'll be back."