European football's governing body said all ties level on aggregate at the end of the second leg would now go to extra-time.
UEFA president Aleksander Ceferin said: "The away goals rule has been an intrinsic part of UEFA competitions since it was introduced in 1965.
"However, the question of its abolition has been debated at various UEFA meetings over the last few years. Although there was no unanimity of views, many coaches, fans and other football stakeholders have questioned its fairness and have expressed a preference for the rule to be abolished.
"The impact of the rule now runs counter to its original purpose as, in fact, it now dissuades home teams - especially in first legs - from attacking, because they fear conceding a goal that would give their opponents a crucial advantage.
"There is also criticism of the unfairness, especially in extra-time, of obliging the home team to score twice when the away team has scored.
"It is fair to say that home advantage is nowadays no longer as significant as it once was.
"Taking into consideration the consistency across Europe in terms of styles of play, and many different factors which have led to a decline in home advantage, the UEFA executive committee has taken the correct decision in adopting the view that it is no longer appropriate for an away goal to carry more weight than one scored at home."
UEFA cited statistics since the mid-1970s which showed how the gap between home and away wins had reduced.
It talked about better pitch quality, standardised pitch sizes, and even VAR as factors in the decline of home advantage.
UEFA said away goals would also no longer be a separating criteria when looking at matches between two or more sides level on points in the group stage.
However, the number of away goals scored in all group matches could be used as an additional separating criteria if required.
Paris St Germain's away-goals victory over Bayern Munich in last season's quarter-finals will go down in history as the last in the Champions League before the rule change.
The rule has led to some dramatic moments in recent years, including Tottenham's stoppage-time success over Ajax in the 2019 Champions League semi-finals.