Baker stuns Monfils in Nice
The 27-year-old qualifier, who is semi-retired and combines playing with a coaching job at Belmont University, broke serve to win the first set before edging a tie-breaker 11-9 in the second.
"I'm very, very excited. This is the first ATP quarter-final I've ever gotten through to," said Baker. "Coming back and being able to play well so quickly, and beat a guy who has been so good, especially on clay, it's amazing.
"I know that I'm not going to beat these guys standing behind the baseline, trying to out-hit them in a 20-ball rally.
"I know I have to step up and be aggressive, use my drop shot and kick serve, and play to my strengths."
Baker's career-high ranking was 172 nearly eight years ago and he is currently ranked 216th after deciding to return to the professional circuit.
He is a wild-card entrant to next week’s French Open at Roland Garros and he continued his form in winning the Savannah Challenger last month by adding French world number 13 Monfils to the scalp of Russia’s Sergiy Stakhovsky.
His win over Stakhovsky was his first on the ATP Tour since 2005, with his last top-level tournament that year's US Open, where he surprised Gaston Gaudio.
Baker next faces Kazakhstan’s Mikhail Kukushkin, who came from a set down to beat Australian fifth seed Bernard Tomic 3-6 6-2 7-5.
Malisse did very little wrong as he remained on terms with Isner, but was outclassed in both breakers as the American claimed a clinical win.
Isner took the opening set 7-3 with a ruthless performance in the breaker after nothing could separate the two players.
The second set was equally tight as neither player faltered on serve, but again Isner had too much for his opponent in the tie-break, taking it 7-5.
Davydenko claimed a 6-4 1-6 6-3 victory with Istomin unable to sustain a high level in the deciding set despite having taken the second comfortably.
The remaining quarter-final will be between France’s second seed Gilles Simon, who beat Grigor Dimitrov of Bulgaria 6-3 6-3m, and Brazil’s Thomaz Bellucci, who earned a 6-4 6-3 win over Spain’s Juan Carlos Ferrero.