Manchester United were far from their best but they did not need to be as Brann gifted them possession on countless occasions and offered very little going forward.
Jess Park saw her effort blocked in the box when United won the ball in midfield, moments before Terland pounced on a loose pass from Ingrid Stenevik to make it 1-0.
Norwegian Terland, who played for Brann between 2021 and 2022, was given too much space on a corner just four minutes later when she peeled off a defender at the back post and nodded in her second goal, putting United ahead on aggregate in the process.
She could have had a hat-trick before the break but curled an effort on to the post as United eased through the game without really imposing themselves.
But her third goal was worth the wait as she ghosted in behind and carried the ball into space, before sliding it past goalkeeper Panengstuen to all but confirm United's aggregate victory.
It caps a strong qualifying campaign for Terland, who has netted seven goals in United's four matches and will hope to play an equally important role in the main draw.
She has been ever present since joining United in 2024, filling the void left by England international Alessia Russo, and they will need her to be at her best against Europe's elite.
Aside from her goals she was intelligent with her movement, tireless out of possession and ruthless when she needed to be.
United, and Terland, will be able to test themselves further against Champions League winners Arsenal on Sunday - the first big test of the WSL season and a marker of the opposition they will have to face now they have accomplished their mission of securing a spot in the main draw.