Alexander-Arnold clipped a pass over Ake's head and on to Salah, who nodded superbly over Begovic and into the net for his 40th goal of a remarkable debut campaign with the Reds.
Liverpool dominated from start to finish on their return to Premier League action, scoring in the seventh minute through Sadio Mane and adding further strikes from Mohamed Salah and Roberto Firmino in the second period.
The victory rounded off a fine week for Liverpool, who booked a Champions League semi-final against Roma by beating Manchester City 5-1 on aggregate with a second-leg win at the Etihad Stadium on Tuesday.
- Sadio Mane is now the outright top scoring Senegalese player in Premier League history, scoring 44 goals in the competition and eclipsing Demba Ba (43).
- Mohamed Salah is the eighth different player to score 30-plus goals in a single Premier League season - he is also the first African to do so. Many football supporters in England are guilty of exaggerating the ability of a highly-rated youngster but in Alexander-Arnold the country - and the Reds - have a real gem.
Yet even that attack was halted by the assistant's flag as Ryan Fraser was offside prior to collecting Charlie Daniels' back-heel and crossing for Jermain Defoe to tap home.
Liverpool: Karius, Alexander-Arnold, van Dijk, Lovren, Robertson, Oxlade-Chamberlain, Henderson (C), Wijnaldum, Salah, Mane and Firmino.
Liverpool welcome captain Jordan Henderson back from suspension for Bournemouth's trip to Anfield.
56' - CLOSE! Van Dijk has been widely praised for his defensive performances of late and now he nearly gets on the score-sheet!
69' - GOAL! Liverpool 2-0 Bournemouth: Salah watches Alexander-Arnold's outrageous pass all of the way and then spectacularly cushions the ball over the head of Begovic. It was his 30 Premier League goals.
Bournemouth boss Eddie Howe admits they failed to match Liverpool's aggression for Saturday's 3-0 defeat.