The Uruguayan directed James Morrison's cross with a flying near-post header 15 minutes from time to cancel out Kevin Mirallas' first-half opener in Pepe Mel's first game.
The Uruguayan scored the first goal of his Premier League career midway through the second half to hand West Brom a point in a game largely dominated by the visitors.
Roberto Martinez's men knew that a win would take them into the top four and boost their hopes of a place in the Champions League.
And the visitors looked to be on the way to a routine victory when Kevin Mirallas opened the scoring just before the interval.
However, Everton was thwarted by a brilliant header from Lugano, which ensured that Mel - who took over earlier this month following the departure of Steve Clarke - started his reign at The Hawthorns on a positive note.
Nicolas Anelka started up front for West Brom on the same day club shirt sponsor Zoopla announced it would not renew its deal with the club in light of the Frenchman's controversial 'quenelle' gesture in the draw with West Ham earlier this month.
Matej Vydra, Lugano, Zoltan Gera and Claudio Yacob also came into Mel's first starting XI, while Sylvain Distin returned from a hamstring strain for Everton, whose bench featured new signing Aiden McGeady.
West Brom made a bright start to proceedings, but it was Everton who created the better chances, midfielder Bryan Oviedo threatening twice early on.
The Costa Rican volleyed wide from the edge of the box with his first effort, then skewed another wide when well placed inside the area.
It took the hosts almost half an hour to craft a clear goalscoring opportunity as Tim Howard was forced into a strong save to deny Zoltan Gera following an excellent pass from Anelka.
But West Brom failed to take advantage of some good spells of possession in the first half, and was made to pay four minutes before the break when Mirallas struck.
The Belgian displayed great awareness to beat Lugano to a Romelu Lukaku knock-down and slot the ball coolly past Ben Foster.
West Brom made a change at halftime, former Everton striker Victor Anichebe replacing Vydra.
That alteration looked to have had little impact on the game, though, as Martinez's men continued to frustrate a home side seemingly devoid of attacking inspiration.
Still, there was to be a twist in the tale on 75 minutes when Lugano turned James Morrison's pinpoint near-post cross into the top corner with a fine header.
Everton thought it had snatched all three points when Distin turned in Gareth Barry's low drive, but the flag was correctly up for offside.
The draw leaves West Brom four points clear of the drop zone, while Everton is sixth, a point behind city rivals Liverpool in fourth.