Peter Bosz's men appeared to be on course to bounce back from last week's surprise defeat at Roda JC when Patrick van Aanholt struck after just six minutes, while Lucas Piazon added their second shortly before half-time.
But Linssen pulled one back in the 51st minute and then earned a share of the spoils in the 77th minute.
Vitesse's failure to win means Ajax could move above them in the standings if they beat Roda on Sunday in the final round of matches before the festive break.
Heerenveen took out their frustrations at being knocked out of the KNVB Beker by AZ with a thumping 5-1 victory at AFAS Stadion.
AZ came out on top on penalties in the cup competition but they were 3-1 down by half-time on Saturday, with Bilal Basacikoglu opening the scoring in the 12th minute before Hakim Ziyech doubled Heerenveen's lead.
Aron Johannsson pulled one back in the 40th minute but Luciano Slagveer restored the visitors' two-goal lead on the stroke of half-time. Alfred Finnbogason made sure there was no way back for AZ in the 69th minute, with Ziyech rubbing salt into the wound four minutes later.
Lex Immers scored twice in seven minutes as Feyenoord moved a point behind Ajax and FC Twente with a routine 3-0 win over PEC Zwolle.
Immers scored in the 33rd and 39th minutes before Jean-Paul Boetius sealed victory for Ronald Koeman's men in the 55th minute.
Anouar Hadouir's 74th-minute effort ensured NAC ended a run of four games without a win after they edged out Cambuur Leeuwarden 1-0.