Daniel Sturridge opened the scoring with a low finish after he was played through by Phillippe Coutinho's flick.
Lovren, a £20 million signing from Southampton, then headed in the second from Steven Gerrard's corner.
Coutinho scored from a narrow angle before Jordan Henderson tapped in a fourth from Sturridge's cross.
As well as Lovren, right-back Javier Manquillo, a loan signing from Atletico Madrid, also had his first outing for the Reds, while new signing Emre Can played in he first half.
Striker Rickie Lambert replaced Sturridge with 15 minutes left for his first appearance at Anfield since joining from Southampton.
Brendan Rodgers' Liverpool, who finished second last season, start their Premier League campaign against Southampton on Sunday.
Dortmund face Bayern Munich in the German Super Cup on Wednesday and start their Bundesliga campaign against Bayer Leverkusen on 23 August.