The Gunners took the lead just past 20 minutes through Tierney, who beat his defender twice, cut into the penalty area and smashed an unstoppable strike past West Brom's Sam Johnstone to put his side in front.
An elegant team move saw Arsenal double their lead minutes later when Emile Smith Rowe squared for Saka to finish off a sequence he helped start on the edge of the attacking third.
Lacazette nearly made it 3-0 just before half-time, but Johnstone stood his ground on the Frenchman's shot from a tight angle to keep West Brom's deficit at two at the break.
Saka was at it again in the second half to help Arsenal find their third goal on 60 minutes. After blowing past his marker, he crossed into the area and following a bizarre sequence of rebounds the ball fell to Lacazette to side-foot home.
"The perfect way to start the New Year, we've won three in a row so we want to keep building. We're really happy with the performance," Saka said.
"It was so cold but I was getting so many touches in good positions, so I enjoyed the game a lot."
Lacazette struck again soon after on a counter-attack when he reacted quickly to redirect a pinpoint cross from Tierney to put Arsenal four goals to the good.
The result sees Arsenal creep up to 11th place in the Premier League while West Brom remain mired in 19th place in the relegation zone.