Nuno was taking charge of his first match as Hammers boss just 48 hours after being hired to replace the sacked Graham Potter.
The former Nottingham Forest manager watched as Michael Keane gave Everton the lead in the first half at the Hill Dickinson Stadium.
England forward Bowen ensured Nuno avoided a debut defeat when he netted after the interval.
It was only the second time in seven games in all competitions this season that West Ham had managed to escape without a loss.
The east London club remain second bottom of the Premier League with four points from their first six games.
“We did a lot of good things. We competed well but there is a lot of work to be done. We started the second half on the front foot, the desire was there,” Nuno said.
“We have to build on that. Now it is about keeping it simple and trying to give the boys a good identity. We will get there,” he added.
With valuable experience of successfully navigating a relegation battle after keeping Forest and Wolves in the top-flight, Nuno has been tasked with guiding West Ham to safety.
The 51-year-old will be encouraged by this battle display after just two training sessions working with his new players since taking charge.
“It is three days, we are in the process of knowing each other. But it’s nice to be back. It’s what we live for – to compete.”
Potter was dismissed on Saturday after only eight months in charge, with the club languishing in 19th place following a 2-1 defeat against Crystal Palace in his last match in charge.
Nuno had been sacked himself earlier this season after falling out with Forest owner Evangelos Marinakis over the club’s transfer policy.
Nuno had masterminded Forest’s return to European competition for the first time in 30 years after an impressive seventh place finish last term. (AFP)
