Everton took the lead on eight minutes in the "mini-derby" at Anfield when Broadhead tapped in Josh Bowler's cross at the back post.
The visitors almost doubled their lead when Lewis Gibson met a Broadhead free-kick but his header hit the bar.
Broadhead sealed the points on 87 minutes, finishing off a pass from Anthony Gordon.
Everton have 37 points, four clear of Brighton & Hove Albion with four matches left.
Liverpool are fifth on 25 points after their first defeat since November.