Everton goalkeeper Jordan Pickford also delivered a confident display as the Three Lions followed up a 5-0 victory over Czech Republic on Friday with another comprehensive success.
Toffees centre-back Keane, winning his seventh senior cap for his country, rose highest to plant a precise header into the bottom corner on the 30-minute mark to get England back on level terms in Podgorica.
Marko Vesovic had given Montenegro a shock lead, curling in an unstoppable effort after 17 minutes.
England responded to the setback in impressive fashion, though, and after Keane made it 1-1, former Everton midfielder Ross Barkley prodded in the away side’s second before the half-time interval.
Barkley’s whipped finish extended England’s advantage on the hour, before Harry Kane tapped home for the Three Lions’ fourth.
Raheem Sterling wrapped up the scoring 10 minutes from time as he raced through on goal before nutmegging Montenegro goalkeeper Danijel Petković.
Cenk Tosun was among the goals as Turkey also made it two wins from two in UEFA EURO 2020 qualifying Group H.
The Everton striker bagged a brace as the Turks beat Moldova 4-0 in Eskisehir.
After Hasan Ali Kaldirim put Senol Gunes’ side in front, Tosun’s powerful left-footed strike doubled their advantage on 26 minutes.
The 27-year-old headed in from Deniz Turuc's corner delivery to further stretch Turkey’s lead eight minutes after the break, before Kaan Ayhaan made it 4-0 as the home team eased to victory.
Two goals in quick succession for Turkey! 🇹🇷
Tosun hammers home from close range to double the hosts' advantage! 🔨 pic.twitter.com/1ek1P3isPk
— Sky Sports Football (@SkyFootball) March 25, 2019
Also in Group H, Everton playmaker Gylfi Sigurdsson started but his Iceland team fell to a 4-0 defeat against France in Paris.
Samuel Umtiti, Olivier Giroud, Kylian Mbappe and Antoine Griezmann netted at the Stade de France as the world champions joined Turkey on six points at the top of the group.
Toffees defender Kurt Zouma was an unused substitute for Les Bleus.
Everton Under-23s striker Fraser Hornby, meanwhile, played the second half as Scotland Under-21s fell to a narrow 2-1 defeat to Sweden in a friendly international.
All the goals came in the first 45 minutes at the Marbella Football Centre, Spain, with Scotland defender Barry Maguire equalising after Joel Asoro’s opener.
Benjamin Mbunga re-established the Swedes’ lead however and, despite the introduction of talented frontman Hornby, Scot Gemmill’s team could not find a leveller.
Everton Under-18s skipper Einar Iversen was also in action in Marbella.
The midfielder completed 90 minutes for Norway Under-21s, but his team were beaten 5-1 by Russia.
The score was 1-1 at the break but four goals in the last 27 minutes saw the Russians run out comfortable winners.