With Tomáš Souček breaking the deadlock midway through the second half, Graham Potter looked set to return from Merseyside with a welcome win.
But Jake O’Brien rescued a last-gasp point for David Moyes with a stoppage-time leveller.
“You always get that feeling of disappointment when you concede so late on but you also have to put it into the context of the overall game,” said Potter.
“If you don’t go on to score that second goal to make it 2-0, then you’re always leaving yourself open to conceding an equaliser.”
During a subdued opening period, Everton keeper Jordan Pickford pulled off two superb stops to prevent a Jarrad Branthwaite own goal before tipping a dipping 20-yard volley from England international teammate Jarrod Bowen over the bar.
Everton had failed to convert the handful of half-chances that came their way, yet moments before the interval, Hammers hearts were in their mouths, when striker Beto hit the deck under pressure from Max Kilman and Konstantinos Mavropanos.
But after pointing to the spot, referee Darren Bond was then directed ro his pitchside monitor, whereupon he reversed his decision to award an Everton penalty to a deafening chorus of Blues boos.
“I felt we restricted them in the first half and Alphonse Areola didn’t have too much to do but we were defending far too deep for my liking,” added Potter, who then saw Souček net for the seventh time this season, on 67 minutes.
“I was pleased to see us come out after the break and score a good goal.”
The 77-cap Czech Republic midfielder, who collected from Bowen and struck a sweet, low curling 15-yarder beyond Pickford, will certainly be sorry to see the Toffees leave Goodison Park, where he has bagged three goals from five visits.
“At 1-0, Everton will always make it tough for you, especially when they’ve got their crowd behind them,” said the Hammers head coach, whose victory hopes were destined to be dashed by O’Brien’s close-range header as the contest entered additional time.
“Goodison Park is as tough as it gets and I’ve got huge respect for the football club and their fans. It’s always been a privilege just to be playing here. We knew that balls would start coming into our box and we ended up conceding. It is what it is.”
Now heading into the international break in 16th spot, Saturday’s result leaves Potter with three wins, two draws and five defeats since taking charge in early January.
“My first ten games have been both enjoyable and challenging,” he said.
“We’ve done some things well and there are still some things that need improvement but it’s pleasing that we’ve taken some big steps in our last four Premier League matches, picking up seven points.”
The Hammers return to Premier League action when they welcome AFC Bournemouth to London Stadium on April 5 (3pm).